# Physics 135 Exam 1

Vol = 8
Area = 4
(2r^1)^2 α 2r^2 = 4A
(2r^1)^3 α 2r^3 = 8V
Doubling the size of the radius increases the volume by a factor of (1) and area by a factor of (2)
The volume increases by factor of 27

volume α size^3 α (diameter^1)^3
V2 α (3d^1)^3
V2 α (3V1)^3
V2 α 27V1

If diameter of a sphere is tripled, what does the volume increase a factor of?
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Mass will increase by factor of 8

M α size^3
M2 α (2size)^3
M2 α 8 size

Size of cow increases isomorphically by factor of 2. What factor will its mass increase?
Mass of miniature breed is predicted to be 209 kg

M1 α L1^3
M2 α L2^3, so…
M1/M2 α L1^3/L2^3
M2 α M1(L2^3/L1^3)
M2 α (1000 kg)(86^3 cm/145^3 cm)
M2 α 209 kg

Typical bull of cattle is 145 cm tall and mass of 1000 kg. What mass do you predict for bull of miniature breed that is 86 cm tall?
Linear with mass (MR α M)

Metabolism α M
Also…
BMR α M^3/4
Heat gained/lost α M^2/3

Energy conversion is driven by mitochondria number. Mitochondria number is independent of size, and number of cells α mass. How does metabolic rate (MR) α mass (M)?
The final circumference is 26.8 cm

C1 α V^1/3
C2 α V^1/3
Take ratio and solve for C2

Statue is scaled down. Size changes without shape change. Starts with V = 4.75 m^3 and ends with V = 1.00 m^3. The initial circumference of a circular base is 45.o cm. What is its final circumference?
The final surface area is 0.354 times the initial surface area

A α V^2/3 <--- [(V^1/3)^2 = V^2/3] A1/A2 α (V1^2/3) / (V2^2/3) Solve for A2

Statue is scaled down. Size changes without shape change. Starts with V = 4.75 m^3 and ends with V = 1.00 m^3. How the final surface area compare to the initial surface area?
Organisms of different sizes accelerate at different rates when jumping. Imagine that we increase your size isomorphically by a factor of 2.2. By what factor does your maximum acceleration change? Hint: muscle cells can apply a fixed stress, so the force muscles can create scales like cross-sectional area.
Knowing that a cow is 5-times as heavy as its calf (baby), figure out a (spherical cow) approximation for the ratio between the skin (hide) area of the cow to the skin area of the calf’s.
We have seen that Kleiber’s Law predicts that the optimal drug dosage scales with mass M as Dose ~ M 3/4 for a wide range of mammals, including people and cattle. A veterinarian wants to give the correct dosage of a drug to a miniature bull with a height of 86 cm but does not have a scale available to weigh the animal. The veterinarian knows the correct dosage for a 1000 kg bull with a height of 145 cm and that bulls scale
isomorphically with size, as we demonstrated in class. The veterinarian therefore scales the dosage for the small bull as Dose ~ ?
After watching King Kong, a grad student in Dr. Meiners’ lab decides to isomorphically resize a gorilla that is 1.5 m tall, so that he becomes 15 m tall. Estimate how much the gorilla weighs before and after resizing. For full credit, make your estimates numerically, based on an assumed shape and
estimated dimensions for a gorilla.
After watching King Kong, a grad student in Dr. Meiners’ lab decides to isomorphically resize a gorilla that is 1.5 m tall, so that he becomes 15 m tall. Will his increased strength be enough to lift and overturn a car? Assume he was able to lift a mass of 90 kg before he was resized, and that muscle strength is proportional to the cross-sectional area of his muscle fibers. For comparison, you can take the mass of a car to be 1200 kg. Show your calculations.
After watching King Kong, a grad student in Dr. Meiners’ lab decides to isomorphically resize a gorilla that is 1.5 m tall, so that he becomes 15 m tall. Would King Kong be able to climb the Empire State building? Consider the stress that King Kong’s leg bones would bear and mathematically defend your
answer. You can assume that in normal activities a gorilla’s bones typically experience no more than 20 % of the stress required to break them.
45 degrees
relative = 1.70 m/s

Vds = Vdw + Vws

V(ds) = tan-1 (1.2/1.2)

Dog swims at 1.2 m/s relative to the water. Water flows left at 1.2 m/s. If dog swims relative to the water with nose pointing to the opposite shore, at what degree will it travel relative to the shore? What is the relative velocity?
60 degrees south of east

cos-1(1/2)

Imagine a dog swimming across a river. The dog can swim at 2 [m/s] relative to the water. The water flows West at 1 [m/s] relative to the shore. What direction must the dog swim to end up directly across the river as shown?
0

A * B = AxBx + AyBy + AzBz
(3)(6) + (9)(-2) + (0)(3)

What is the dot product of (3x + 9y) * (6x – 2y + 3z)
6x – 9y

A * B = (AyBz – AzBy)x + (AzBx – AxBz)y + (AxBy – AyBx)z
[(2)(3) – (0)(0) * x] + [(0)(0) – (3)(3) * y] +
[(3)(0) – (2)(0) *z]

Cross product of (3x + 2y) * 3z
7.1 m/s

NE is exactly 45 degrees
Vy = 10 * cos(45)

A falcon flies east with a speed of 10 m/s Northeast. How rapidly is it moving East?
2.24 m/s
Car moves towards blackboard towards y-direction with speed of 2m/s relative to the sheet. The sheet moves to the right relative to the bench at speed of 1 m/s. What is the speed of the car relative to the bench?
63.4 degrees

tan-1(2/1)

Car moves towards blackboard towards y-direction with speed of 2m/s relative to the sheet. The sheet moves to the right relative to the bench at speed of 1 m/s. What is the angle of the car’s velocity vector relative to the bench make with in the x-direction?
North: 3043.4 miles
3200 * cos(18)

East: 988.85 miles
3200 * sin(18)

A bird migrating towards summer nesting site travels 3200 miles in a direction 18 degrees East of North. How much farther North is the bird at the end of this migration? How much farther East?
10.54 degrees

tan-1(800/4300)

A bird travels 4300 miles North and 800 miles West to Alaska. What direction does it travel as an angle away from true North?
(-11,14,8)

-2A: -2x + 0t + 6z
3B: -6x+15y+3z
-C: -3x-1y-1z

V(A) = (1,0,-3)
V(B) = (-2,5,1)
V(C) = (3,1,1)

Find V(-2A) + V(3B) – V(C)

(17,-12,-6)
V(A) = (1,0,-3)
V(B) = (-2,5,1)
V(C) = (3,1,1)

Find V(2A) – 3[V(B) – V(C)]

cos(theta) = AxBx + AyBy + AzBz / sqroot[(Ax)^2 + (Ay)^2 + (Az)^2] * sqroot[(Bx)^2 + (By)^2 +(Bz)^2] cos (theta) = -0.690065559
theta = Cos^-1 (-0.69006559)

A: (2,1,-4)
B: (-3,0,1)
C: (-1,-1,2)
Scalar Product = -10 from A * B

What is the angle between A & B in radians (1 sigfig)?

V(mf) = 5.0 m/s
to the right

V(mf) = V(mo) – V(0f)

A railroad flatcar traveling to the right at a speed of 13.0 m/s relative to the observer standing on ground. Someone is riding a motor scooter on the flatcar. What is the magnitude of velocity of motor scooter relative to the flatcar if its velocity relative to observer on ground is 18.0 m/s to the right? What is the direction of the velocity of the motor scooter relative to the flatcar?
16.0 m/s
to the left

V(mf) = V(mo) + V(of)

A railroad flatcar traveling to the right at a speed of 13.0 m/s relative to the observer standing on ground. Someone is riding a motor scooter on the flatcar. What is the magnitude of V(mf) is V(mo) is 3.0 m/s to the left? What direction is V(mf)?
13 m/s
to the left

V(mf) = V(mo) – V(of)

A railroad flatcar traveling to the right at a speed of 13.0 m/s relative to the observer standing on ground. Someone is riding a motor scooter on the flatcar. What is magnitude of V(mf) if V(mo) = 0? Direction of V(mf)?
T = (L/V0+V) * S

T = (L/V-Vo) * S

The sidewalk in an airport building moves at speed of V0 and is the length of L. Women steps on at one end and walks at a speed of v relative to the moving sidewalk. How much time does she require to walk from one end to the other if she walks in same direction the sidewalk is moving? If she walks opposite to the direction the sidewalk is moving?
10. 8 degrees North of West

tan^-1(80/425)

An airport pilot flies due West. Wind of 80.0 km/h blows South. If airspeed of plane (speed in still air) is 425 km/hr, what direction should pilot head? Express as angle measured North of West.
417 km/hr

V(ac) = sqroot [(Vab)^2 – (Vbc)^2]

An airport pilot flies due West. Wind of 80.0 km/h blows South. What is the speed of the plane over ground?
Vector A has a magnitude 5.00 and points in a direction 40.0º clockwise from the negative yaxis. What are the x and y components of vector A ?
An airplane undergoes the following displacements: First, it flies 66 [km] in a direction 30º east of north. Next, it flies 49 [km] due south. Finally, it flies 100 km 30º north of west. Using vector components, determine how far the airplane ends up from its starting point.
Determine the scalar product
A = 6.0xˆ + 4.0yˆ − 2.0zˆ
B = 5.0xˆ − 6.0yˆ − 3.0zˆ
What is the angle between the vector A = 3xˆ − 2yˆ − 3z and the positive y-axis?
What is the cross product
A = 2xˆ + 3yˆ +1zˆ
B =1xˆ − 3yˆ − 2z
A 30 [m] wide river flows from east to west at a constant speed of 1.2 [m·s−1]. A boat steers straight north, directly across this river, traveling with a speed of 2.2 [m·s−1] relative to the water. If the boat wishes to travel directly across the river (straight north), in what direction should the pilot steer?
Calculate the magnitude of the vector difference using the information provided in the drawing below (look at practice exam question #3)
A dog wishes to cross a stream that is 20 [m] wide and runs from West to East. The stream water flows at 0.5 [m/s] East. If he points himself South, swimming at 1 [m/s] with respect to the water how far East does the dog travel by the time he crosses the river?
Airplane One flies due east at 250 km/h relative to the ground. At the same time, Airplane Two flies 325 km/h, 35° north of east relative to the ground. What is the velocity of Airplane One relative to Airplane Two?
If C = -5x -2y-3z, What is C * z?
Popeye wishes to row across a straight stretch of river to a point on the other shore that is directly opposite his starting point, where Olive Oyl awaits him. He regularly rows at
2.0 m/s in still water, and the river is flowing parallel to the shore at 0.48 m/s toward the left side of the page. At what angle with respect to the shoreline should Popeye steer his
rowboat in order to make his desired landing?
If V(A) = 3x + 4y , then a unit vector A pointing in the same direction as V(A) is given by…
A vector F1 has a magnitude of 40.0 units and points 35.0° above the positive x axis. A second vector F2 has a magnitude of 65.0 units and points in the negative y direction. Use the component method of vector addition to find the magnitude and direction, relative to the positive x axis, of the resultant F = F1 + F2.
Two vectors have nonzero magnitudes. Their dot product is zero. Which of the
following statements are true?
I. The vectors are parallel.
II. The two vectors are perpendicular.
III. The two vectors have a zero cross product.
IV. The two vectors have a nonzero cross product
Consider two vectors A= 3x+2y-z and B = 2x – 4y -3z
Calculate A * B
Calculate |A| and |B|
Calculate the angle in degrees between A and B
At what time are the two trains accelerating along their direction of motion? Put another way, at what time are both trains speeding up in whatever direction they are going? (Look at practice exam question #1)
A motorboat has a still water speed of 7.5 m/s. The driver of the boat steers a course due north across a river which flows at 3.0 m/s due east. If the river is 520 m wide, how long does it take the boat to cross the river?
How far would you have to fall to have the same momentum as you do when you are traveling down the road at 70 mph in your car? (Remember that 1 m/s ≈ 2.24mph). Ignore the effects of air friction for this calculation.
– Bob will exit vehicle forwards and Floyd will remain in vehicle
– The vehicle stopped and Bob continued to move to the right
– Seat belt and shoulder harness pushing on Floyd to the left caused him to slow down with the vehicle
– Bob’s exit = 1st law
– Floyd’s remaining in vehicle = 2nd law
In a car, Floyd is in the back seat and has a fastened seat belt and shoulder harness. Bob in the front seat does not. Vehicle moves to the right and experiences a sudden stop. The vehicle is moving at a constant, high speed before it suddenly stops. What will happen? Why? What Newton Law accounts for this/these situation(s)?
No
Would Bob’s weight downward and upward normal force on his posterior from seat be an action-related pair?
No
Can an airplane in free flight turn in a horizontal circle with its wings level?
The weight must act downward. There must be a component of force to the right to change the direction of the momentum vector toward the circle center. Note that the
horizontal component of the lift changes the direction
of the momentum while the vertical component
supports the weight of the airplane.
Draw a correct free body diagram for an airplane turning in a circle centered to the right of the page
W = weight
L = lift
D

There is a net force, so the object’s momentum vector must change with time. But we have no information about the direction of the force relative to the momentum. As a result, the object could be changing speed only (force parallel to momentum or object at rest), changing direction only (force perpendicular to momentum) or both (force with components both parallel and perpendicular to momentum). Without
information about the directions of the force and momentum vectors we cannot distinguish between these possibilities.

An object has one force acting on it. From this
we can conclude with certainty that
A. It is changing speed.
B. It is changing direction.
C. It is changing speed and direction.
D. None of these.
1) Fground on you (normal)
Ground pushes up on you, you push down on ground

2) Fground on you (friction)
Ground pushes forward on you, you push backward on ground

3) Fgravity on you
Earth pulls down on you, you pull up on Earth

4) Fair on you
Air pushes on you, you push on air

You are riding your bike, peddling steadily and traveling at a constant speed to the right. —>’s represent forces.

1) What force does —>(1) pointing upward from bike represent?
2) What force does —> (2) pointing from bike to the right represent?
3) What force does —> (3) pointing downwards from bike represent?
4) What force does —> (4) pointing to the left from the bike represent?

ALSO, what is a 3rd law partner of each one? (All act on you)

equal to 0
Car moves to left with constant velocity. The net force on the car is…

a. greater than 0
b. equal to 0
c. less than 0

equal to 0
Object can’t remain at rest unless the net force on it is…

a. greater than 0
b. equal to 0
c. less than 0

1) T
2) T
3) F
4) F
5) T
6) F —- sum of forces = m * a
True or False

1) Every force has one and ONLY one 3rd law pair force
2) 2 forces in each pair act in OPPOSITE directions
3) 2 forces in each pair can either both act one SAME body OR on DIFFERENT bodies
4) 2 forces in each pair may have different origin (Ex. one force due to gravity and its pair force is due to friction)
5) 2 forces of 3rd law pair ALWAYS act on different bodies
6) Given 2 bodies interact via force, accelerations of the 2 bodies have EQUAL MAGNITUDE but OPPOSITE DIRECTIONS. NO other forces act on either

magnitude
antiparallel (opposite)
the moon
Force on small moon due to large Earth is equal in _______ but ______ to force on Earth due to ________.
1) gravity
2) 5N; book
3) No
4) 5N; Earth; book; upward
5) 5N; table; book; downward
6) 1st and 2nd
7) 3rd
Book weighs 5N and rests on top of table.

1) Downward force of magnitude 5N is exerted on book by which force
2) Upward force of magnitude of ____N is exerted on ______ by table.
3) Does downward force and upward force from 1 & 2 constitute a 3rd law pair?
4) Reaction to force by 1 is force of magnitude ____N exerted on the ____ by the _____. Direction is _______.
5) Reaction to force by 2 is force of magnitude ____N exerted on the ____ by the _____. Direction is _______.
6) Which law predicts that the forces in 1 & 2 are equal and opposite?
7) Which law predicts that forces in 2 & 5 are equal and opposite?

2 * Fab *Cos(theta)
Six bollards are available to tie boat at dock. She has 3 ropes. She can tie boat’s center (A) to any bollard (B-G). Suppose one runs to A from B, 2 to A from D & 3 from A to F.
F(AB) = force on boat from rope running to A from B
Fab = magnitude of that force

What is the magnitude of the force provided by the 3rd rope in terms of the angle?

* Look at figure in practice exam, number 4*
Three forces are exerted on an object placed on a tilted floor. Forces are vectors. The three forces are directed as shown in the figure. If the forces have magnitudes F1 =1.0 [N], F2 =8.0 [N] and F3 = 7.0 [N], where [N] is the standard unit of force, what is the component of the net force parallel to the tilted floor?
The International Space Station has a mass of 1.8 × 105 [kg]. A 70.0 [kg] astronaut inside the station pushes off one wall of the station so she accelerates at 1.50 [m·s−2]. What is the magnitude of the acceleration of the space station as the astronaut is pushing off the wall? Give your answer relative to an observer who is space walking and therefore does not accelerate with the space station due to the push.
A very light 1.00 [m] wire consists of two segments of equal length, one of steel (Young’s modulus is 2.00 × 1011 [N·m−2]) and one of brass (Young’s modulus is 9.0 × 1010 [N·m−2]).
The steel segment is 1.50 [mm] in diameter, and the brass segment has twice this diameter. When a weight W is hung from the ceiling by this wire, the steel segment stretches by 1.10 [mm]. Find the weight W.
A ship is being pulled through a harbor at constant velocity by two tugboats as shown in the figure. The lines attached to the two tugboats have the same tension of 2 × 105 [N]. Each line
makes an angle of 28º with the direction the ship is being towed. What is the magnitude of the drag force due to the water on the ship?
Madeleine drives her car down the street. She gently applies the brakes to stop at a light. Which of the following forces are NOT “action-reaction” (or 3rd−law) pairs?

(A) The force the seat belt and shoulder harness exert on Madeleine and the force Madeleine exerts on the seat belt and shoulder harness.
(B) The downward force that Madeleine exerts on her seat and the upward force applied to her by the seat.
(C) Madeleine’s weight downward and the upward normal force applied to her by the seat.
(D) The downward force that the car exerts on the road and the upward normal force that the road exerts on the car.
(E) All of the above are action reaction forces.

Madeleine is on an elevator that is accelerating downwards at 2 [m/s2]. If her mass is 55 [kg], what is the normal force applied to her by the elevator?
A child slides down an icy incline of 30°. Ignoring frictional forces, if the child begins at rest how long does it take her to travel a distance of 20 [m] along the incline to the bottom? The
combined mass of the child and sled is 10 [kg].

What graph best describes the child’s horizontal position as a function of time? Note that the horizontal direction is along the x axis, with movement to the right as positive.

You are standing on a 100 [m] cliff. You throw a ball upward with a vertical speed of 3 [m/s]. How long does it take for the ball to reach the ground 100 [m] below?
A toboggan loaded with students of total mass 500 [kg] slides down a snow-covered slope with constant velocity. The hill slopes at a constant angle θ, and the coefficient of kinetic friction µk between the toboggan and the snow-covered slope is 0.58. Find the the angle θ of the slope.
My copy of The Complete Works of Shakespeare has a mass of 3.5 [kg] hold the book up against the wall by pushing it with a force of 50 [N] at an angle 60° above the horizontal. The book doesn’t move. What is the magnitude and direction of the frictional force exerted by the wall on the book?
A horse pulls a cart along a flat road. Consider the following four forces that arise in this situation.
(1) The force of the horse pulling on the cart
(2) The force of the cart pulling on the horse
(3) The force of the horse pushing on the road
(4) The force of the road pushing on the horse

Which two forces form an “action-reaction” pair that obeys Newton’s Third Law?
A) 1 and 4
B) 1 and 3
C) 2 and 4
D) 3 and 4
E) 2 and 3

The following graph (number 9 in practice exam) shows the velocity vs. time for a particle moving along the x−axis.

Which statement is true?
A) The particle is farthest from its starting point at D.
B) The acceleration is greatest at B.
C) The particle is moving in the opposite direction at C than it is at A.
D) The particle is closer to its starting point at C than it is at B.
E) The particle is momentarily at rest at B.

A hockey puck initially moving with a speed of 2 m/s slides to a stop on the ice in a distance of 10 m. What is the magnitude of the puck’s acceleration as it slows and stops? You may assume a constant acceleration.
A block just slides when placed on an inclined plane with an angle (c)= 33 degrees. The coefficient of static friction between the block and the inclined plane is therefore . . .
A boy pulls a sled of mass 5.0 kg with a rope that makes a 60.0° angle with respect to the horizontal surface of a frozen pond. The boy pulls on the rope with a force of 10.0 N and the sled moves with constant velocity. What is the coefficient of friction
between the sled and the ice?
The Starship Enterprise fires its impulse engines to accelerate through the Romulan neutral zone. The thrust from the engines produces a constant force F on the ship. Which of the following statements is true?

A) The Enterprise gains an equal amount of momentum each second
B) The Enterprise gains an equal amount of kinetic energy each second
C) The Enterprise gains an equal amount of momentum for each kilometer through which it travels
D) The Enterprise gains an amount of kinetic energy equal to the amount of impulse provided by the engines
E) The Enterprise gains an amount of momentum equal to the amount of work done by the impulse engines

Which of the following is not an action-reaction pair in accordance with Newton’s Third Law?

A) A horse pulls a cart to the right, the cart pulls the horse to the left.
B) A book pushes down on a table, the table pushes up on the book.
C) The Physics Team pulls the History Team to the right, the History Team pulls the Physics team to the left.
D) Your weight pulling down on you and the floor pushing up on your feet.
E) Your foot pushing on the floor and the floor pushing on your foot as you walk.

You want to move a 250 kg safe down a ramp from the back of a truck at constant speed. The ramp makes an angle of 30° with the horizontal, and the coefficient of kinetic friction is μk = 0.4. What is the approximate magnitude and direction of the force you must apply while it is sliding?
The first figure (practice exam, number 8) below shows a position-time graph, where position is measured in meters and time in seconds. Which of the five proposed velocity-time graphs correctly represents this motion? On all of these, velocity is measured in meters per second and time in seconds.
The force F pushes two blocks across a surface at a constant velocity of 0.54 m/s as shown in the picture. The block masses are m1 = 5.3 kg and m2 = 2.5 kg. If the coefficient of friction between the blocks and the surface is μk = 0.35, what is the size of the force with which block one pushes on block two?
A 10 kg block is at rest on a horizontal surface. A force of 30.0 N is applied to the block at an angle 30° above the horizontal as shown in the picture. Frictional forces prevent the block from accelerating. What is the minimum coefficient of static friction
between the block and the surface which will allow this to occur?
No
Are normal force and weight 3rd law partners?
N = -10Ny

Fy = +30N – 20N – FN = 0 <-- direction determines signs Fn = 10N downwards

You hold a boon of W = -20N(y) against ceiling by pushing on it upwards with force of +30N(y). What is the normal force exerted by ceiling on book?
NF of table on book (+) <---> NF of book on table (-)
GF of Earth on book (-) <---> GF of book on Earth (+)
GF of Earth on table (-) <---> GF of table on Earth (+)
NF of floor on table (+) <---> NF of table on floor (-)
List the 3rd law pairs for a book resting on a table which is resting on the ground and their direction.
T = Wperson = mg

T = [2 (m1 * m2)/ m1 + m2] * g —– system will move but tension is same throughout but with a different value than before

You have two pulleys with one exact clone of a person on each side. What’s the T of the rope? What if one was heavier than the other?
T = 500 N

w = 100kg * 10 m/s
w = 1000 N/2

100 kg piano. Two pulleys. One end of rope attached to ceiling and the other end is being pulled down by a person. How much force is he pulling?
g = 10 m/s
T = 25

100 = 4T

100 lb weight connected to a bar connected to two pulleys. Two more pulleys are connected to a block. One end of the rope is connected to the ceiling while the other end is pulled down. What is T?
T = 1/2(Mwasher + Mplatform) * g

SumF(y) = 2T – (Mwash + Mplat) * g = 0

Washer is on platform which is connected to a pulley. How much should the washer pull down to hold herself in place?
MA = 8

T = w/8
MA = w/(w/8)

Find the mechanical advantage (MA) of a system with 8 tensions on a weight with a mass of W
1) T = Mg

2) T = 1/2 Mg

3) T = 0

A vine with a mass (M) hangs from a tall canopy tree

1) T of vine at point where it connects to tree. Express in terms of M and g

2) What is T 1/2 way down the vine?

3) What is T at the bottom of the vine?

1) T = 36.0 N (T = W)

2) T = 72.0 N (T = 2W)

Two masses of weight 36.0 N are suspended at opposite ends of rope that passes over a light, frictionless pulley. Pully is attached to chain that goes to the ceiling.

1) Tension in rope

2) Tension in chain

T = 2540 N

W = 9.26 x 10^2
T = W
T = Vy
2T * sin(theta) = Vy
solve for T

2 rock cliffs. Person crosses along rope between them and stops at rest at middle of rope. The rope will break if tension exceeds 2.40 x 10^4 N. Mass of person = 94.4 kg. If angle between rope and horizontal is 10.5 degrees, what is the tension in the rope?
1.11 degrees

W = 9.26 x 10^2/2 = 463 N
sin^-1(W/Tmax)

2 rock cliffs. Person crosses along rope between them and stops at rest at middle of rope. The rope will break if tension exceeds 2.40 x 10^4 N. Mass of person = 94.4 kg. What is the smallest value (theta) can have if the rope is not to break?
F = 4150 N

F = mg * sin(theta)

Certain streets make an angle of 18 degrees with the horizontal. What force parallel to street surface is required to keep car with mass of 1370 kg from rolling down the street?
Fman = 283 N
—> F = mg * sin(theta)

Fman = 288 N
—> F = mg * sin(theta) / cos(theta)

Man pushes piano with mass of 160 kg. Slides at constant velocity down ramp inclines at 10.4 degrees above horizontal floor. Neglect friction on piano. What is the magnitude of the force by man pushing parallel to incline? Magnitude if pushed parallel to the floor?
Two weights are connected by a massless wire and pulled upward with a constant speed of 1.50 [m·s−1] by a vertical force F. The tension in the wire is T (see figure). F is closest to what?
A traffic light weighing 100 [N] is supported by two ropes as shown in the figure (number 10). Find the tensions TL and TR in the ropes.
A block of mass M hangs at rest as shown below. The rope that is fastened to the wall is horizontal and has a tension off 52 [N]. The rope that is fastened to the ceiling has a tension of 91
[N], and makes an angle with the ceiling? What is the angle?
Calculate the weight of the crate, in the system of pulleys below, knowing that is lifted with constant speed by a rope-pulling force of 60 [N].
A heavy crate of mass M hangs from a cylindrical steel cable of length L and diameter D as shown below, stretching the cable by a distance d. The steel cable has negligible mass relative to
the mass of the crate. If a second cable is substituted which has twice the length and twice the diameter of the original cable, the new cable stretches by an amount of what?
A 1200 [kg] car is held in place by a light cable on a very smooth frictionless ramp as shown. The cable makes an angle of 31.0° above the surface of the ramp, and the ramp itself rises at 25.0° above the horizontal. How hard does the surface of the ramp push on the car?
A heavy crate of mass M hangs from a cylindrical steel cable of length L and diameter D as shown below, stretching the cable by a distance d. If a second cable is substituted which has twice the length and twice the diameter of the original cable, the new cable stretches by an amount of what?
A gondola car of mass m is temporarily stationary and is suspended at the halfway point between two supports. The cable sags at an angle below the horizontal as shown in
the diagram. Ignoring the mass of the cable, the tension in the cable is what?
A spider of mass ms is hanging from its web as shown below: (number 18)

What is the tension in the thread labeled 2?

For the arrangement of pulleys shown (number 20), with what force F must you pull on the rope to support the 100 kg crate?
A block of mass m1 rests on a frictionless ramp, which makes an angle with respect to the horizontal. It is attached to an ideal massless string that wraps over a massless pulley. A block of mass m2 (m2 < m1) hangs vertically from the other end of the string. For some given values of m1 and m2, what value for (theta) is required for the masses to remain stationary?
Sally is using a block and tackle to lift a box with a weight of 1000 N. She finds that she can lift the box at a constant speed if she pulls on the rope with a tension of 167 N. The number of pulleys attached to the box is therefore what?
A block and tackle is an arrangement of frictionless pulleys and massless ropes something like that shown in the picture. Given this arrangement, what must the tension in the rope be in order to lift the 185 kg mass at a constant velocity?
A 10 kg block is at rest on a horizontal surface. A force of 30.0 N is applied to the block at an angle 30° above the horizontal as shown in the picture. Frictional forces prevent the block from accelerating. What is the minimum coefficient of static friction
between the block and the surface which will allow this to occur?
T increases

L2 = L1
Tv * L2 – mg * L1 = 0
Tv = (L1/L2) * mg
Tv = mg when L1 = L2

A cable supports a hinged rod. As (theta) decreases, what happens to tension? **discussion problem 1**

What value of L2 will vertical component of tension equal the weight of beam?

700 N

Tv * L2 – mg * L1 = 0
Tv = T sin(theta)
T = (mg * L1)/(L2 sin(theta))

Dancer’s outstretched arm is a “uniform rod” hinged at shoulder with weight of 30 N and length of 1 m. Shoulder muscle is cable which attaches to “rod” at distance L2 = 0.1 m and angle of 12 degrees relative to the horizontal. What tension force must shoulder muscle exert to support arm parallel to floor?
M = 800 N

sumtorc = +M * 4 cm – H * 14 cm – w * 30 cm = 0
solve for M

Lifting weight of 100 N. Arm is 30 cm long, weight is 15 N. center of gravity is 14 cm away from elbow. How much force acts on biceps which is orthogonality attached 4cm away from elbow?
M = 800 N

torc = rF(perp)
sumtorc = +M * 4cm*cos(60) – H * 14cm*cos(60) –
W * 30cm*cos(60)
solve for M

Holding 100N. Angle is 60 degrees. Weight of arm is 15 N. What is force on biceps?
L/4

find torcs
1) 1/4w(L-x) – 3/4w(x) = 0
solve for (L-x)
plug and chug

2) find Wperson
Wright[L] – Wperson[x] – Wboard[L/2] = 0
x = L [(Wr-Wb/2)/(Wp)]

Board is 100 N and length is L. Scale on left reads 500 N and scale on right reads 200 N. How far from left end is center of mass of person?

(show both ways to solve this)

distance spring stretches
The force required to change length of ideal spring is proportional to what?
0.75

Center (x) = sum (mx)/m
= (5)(0) + (15)(1)/5 +15

5Kg and 15Kg of weight are connected by 1m long massless bar. If 5kg weiht sits at x=0 and the 15kg weight at x =1m, what is the position of the center-of-mass of this combination?
1) same

2) standing

Consider a mattress made of springs

1) Will Bill impart larger FORCE on mattress if he stands vertically or lying down?

2) Which situation will springs compress more?

1) 300 N

2) 0

3) 150 N

4) 150 N

Uniform 300 N trapdoor in ceiling is hinged at one side.

1) Find net upward force needed to begin to open it, if upward force is applied at center

2) Find total force on door by hinges if upward force is applied at center

3) Find net upward force needed to begin opening if force is applied at edge opposite of hinge

4) Find total force on door by hinges if upward force is applied at edge opposite of hinges

In the figure, the horizontal lower arm has a mass of 2.8 [kg] and its center of gravity is 12 [cm] from the elbow joint pivot. How much force FM must the vertical extensor muscle in the
upper arm exert on the lower arm to hold a 7.5 [kg] shot put?
A 30.0 [kg] child sits on one end of a long uniform beam having a mass of 20.0 [kg], and a 40.0 [kg] child sits on the other end. The beam balances when a fulcrum is placed below the beam a distance 1.10 [m] from the 30.0 [kg] child. How long is the beam?
A block of mass M hangs at rest as shown below. The rope that is fastened to the wall is horizontal and has a tension off 52 [N]. The rope that is fastened to the ceiling has a tension of 91
[N], and makes an angle with the ceiling. What is the angle?
A very light 1.00 [m] wire consists of two segments of equal length, one of steel (Young’s modulus is 2.00 × 1011 [N·m−2]) and one of brass (Young’s modulus is 9.0 × 1010 [N·m−2]).
The steel segment is 1.50 [mm] in diameter, and the brass segment has twice this diameter. When a weight W is hung from the ceiling by this wire, the steel segment stretches by 1.10 [mm]. Find the weight W.
A store sign is suspended in the fashion shown in the diagram. The uniform pole has mass of 4.00 [kg] and is hinged at the point where it is attached to the wall. The horizontal rope is attached to the end of the pole so that the pole makes an angle θ = 35º with the horizontal. When
the sign is suspended from the end of the pole, the tension in the horizontal rope is 150 [N]. What is the mass of the sign? The rope has negligible mass relative to the mass of the sign.
A uniform sign is supported against a wall at point P as shown in the figure. If the sign is a square 0.40 [m] on a side and its mass is 4.0 [kg], what is the magnitude of the horizontal force
that the wall at P experiences? Note that the gap distance between the sign and the wall is negligible relative to the size of the sign.
An 82.0 [kg] diver stands at the edge of a uniform diving board of length 5.00 [m] and mass 10.0 [kg], which is supported by two narrow pillars 1.60 [m] apart, as shown in the figure. Find the magnitude and direction of the force exerted on the diving board by pillar A.
Calculate the tension in the horizontal chain (T2) in the picture below knowing that the weight of the car engine is 500 [N] and both the ring and the chains have negligible masses
You place a brick on top of two identical springs in parallel (one spring right next to the other) and measure the (compression) displacement ∆xparallel of the system. Then you put the two springs on top of each other (configuration in series) and again measure the total displacement of the system when the brick was laid on top of the upper spring; this time you get ∆xseries. The springs have negligible mass relative to the mass of the brick. What is the ratio ∆xseries/∆xparallel?
Newton and Young wish to play on a teeter-totter. If Newton weighs 1.5 times as much as Young what arrangement/s below would produce equal and opposite torque with respect to the
pivot point? Assume that the teeter-totter is well balanced (the center-of-mass of the plank is located at the pivot point).

(A) Young should be seated 1.5 times farther from the pivot point than Newton
(B) Newton should be seated 1.5 times farther from the pivot point than Young
(C) Newton should be seated 0.67 times closer to the pivot point than Young
(D) None of the above
(E) A and C

A man weighing 350 [N] lies on a 1 [m] long board that rests on two scales. The man’s center-of-gravity (CG) is located 0.3 [m] from the left side of the board as shown. If the left scale
reads 300 [N] how much does the right scale read (ie – what is Wright)?
In the figure below, a 10 [m] long bar is attached by a frictionless hinge to a wall and held horizontal by a rope that makes an angle θ of 60° with the bar. The bar is uniform with mass 15 [kg]. How far from the hinge should a 10 [kg] mass be suspended for the tension T in the rope to be 160 [N]?
A machinist is using a wrench to loosen a nut as shown. The wrench is L = 0.25 [m] long, and he exerts a force F = 500 [N] at the end of the handle at θ = 35º with the handle. What is the torque that the machinist exerts about the center of the nut?
A 7-kg block is connected by means of a massless rope and a frictionless, massless pulley, to a second block of mass m hanging on the rope as shown in the figure. The coefficient of static friction between the 7 kg block and the surface it rests on is u(s) = 0.4. What is the largest possible mass m such that the two blocks do not slide?
Consider the diagram below which schematically represents the behavior of natural rubber as it is stretched. The vertical dotted lines divide the graph into three regions A, B and C respectively:

Which of the following statements are true?
I. Hooke’s Law reasonably describes the behavior of the rubber in region B.
II. The rubber is on average stiffer in region A than in region B.
III. The average stiffness of the rubber in region B is less than in region C.
IV. The rubber is stiffer near the origin than in region B.

A weight is placed on top of a spring which compresses a distance delta(x) to support the weight in equilibrium. If instead I now stack three springs identical to the first one on
top of each other and place the same weight on top of them, what distance will the three stacked springs compress to support the weight? You may neglect the mass of the springs.
In the diagram below, both the wrench and the applied force F lie in the plane of the paper. When the force F, with magnitude 80N, is applied to the wrench as shown, the torque produced at point P is what?
A stoplight with a mass of 24 kg is hung vertically from a steel wire with a crossectional area of 3.0 x 10^-6 m^2 and an unstretched length of 2.6 m. By what amount does the wire stretch? The Young’s modulus of steel is 2.0 x 10^11 N/m^2
A mass M placed on top of a spring compresses it spring by a distance x. A second identical spring is placed on top of the first spring and a mass of M/2 is placed on top of the two stacked springs. The two stacked springs are now compressed by a distance of what?
Consider a force F = 3Nx – 4Ny acting at a position r = lmx-2my from a center of rotation on an object. Calculate the resulting torque about the rotation center.
Consider a force F = 3Nx – 4Ny acting at a position r = lmx-2my from a center of rotation on an object. Assuming both r and F lie in the plane of the page, describe the direction of the resulting torque.
Consider a force F = 3Nx – 4Ny acting at a position r = lmx-2my from a center of rotation on an object. Two vectors A and B lie in the page, have equal magnitudes and have a zero cross product: A x B = 0. Assuming A points to the right side of the page, draw the two possible orientations that these vectors can have. Depict each vector as an arrow and label each vector in your two drawings.
Consider a force F = 3Nx – 4Ny acting at a position r = lmx-2my from a center of rotation on an object. Two unit vectors u and v satisfy the condition |u * x| = 1/2. Calculate the
angle in degrees between u and v.
A person lies on a 4m long platform which has a scale under either end. The scale on the left reads 620 N and the scale on the right reads 180 N. When the person gets off the board, each scale reads 50 N.

A. What is the mass of the person?
B. Draw and label all the forces which act on the board not already shown
in the diagram.
C. How far from the left hand end of the platform is the center of gravity of the person?

Picture an 8 m long branch extending straight out from the trunk of a mighty oak. The branch has a mass of 1200 kg and its center of mass is located 3 m from where it attaches
to the trunk. The diameter of the branch is 0.5 m where it attaches to the trunk. To resist the torque this weight generates, the branch is pushed away from the trunk below its
center and pulled toward it above its center. We can model this as in the picture below. What is the magnitude of the force labeled Ftop?
Our old friend the gecko stands on a vertical wall patiently waiting for a tasty insect to wander by. Her mass is M, her center of mass (labeled CG here) is a distance LCM from
the wall and LCG-FF below her front feet, while the distance between her front and back feet is LBF-FF.

a) What is the magnitude and direction of the total
vertical force applied on her feet by the wall.
b) What is the magnitude and direction of the total
force exerted by all of her feet on the wall?
c) What is the magnitude and direction of the
horizontal force exerted by the wall on her back feet (the
ones on the bottom)?
d) Explain briefly what is special about gecko feet that enables them to walk up walls.

Consider static friction between dry solid surfaces. Explain in a few sentences how the coefficient of static friction is related to the magnitude of the static frictional force between two objects.
Imagine the friction between a sneaker with mass msneaker and the slope it rests on is well described by a model in which max 2 F F static N = μs . What is the maximum angle θ the slope could have before the shoe would begin to slip down it?
Fluid friction is a major problem for trees. A horizontal wind provides a stress on the tree which can break its trunk or pull its roots out of the ground. As a result, many leaves change shape when the wind blows, folding over, and reducing
the area they present to the wind. How would you expect this changing shape to affect drag on the leaf? Will this drag force still increase as v2 ? Will it increase more rapidly, more slowly? Describe what you expect to have happen and why in a few sentences.
Streamlining can dramatically reduce the drag associated with fluid friction. Imagine that a cylindrical submarine, with drag coefficient cylinder C = 0.87 , can travel at a maximum speed of 25 m/s. If the submarine kept the same cross-sectional area A, but is streamlined instead, so that its drag coefficient is reduced to streamlined C = 0.08, what would its maximum speed be? Assume that the forward force pushing the submarine through the water remains the same.
Red Kangaroos are quite large, with masses of around 80 kg. They also have very extended feet, around 30 cm in length. Consider the forces which act on such feet when the kangaroo stands at rest. We will ignore the weight of the feet themselves, because they are small compared to the loads they support.
Consider just the right leg, one of the two on which the kangaroo stands. What is the magnitude of the normal force FN which acts on this one leg?
Red Kangaroos are quite large, with masses of around 80 kg. They also have very extended feet, around 30 cm in length. Consider the forces which act on such feet when the kangaroo stands at rest. We will ignore the weight of the feet themselves, because they are small compared to the loads they support. If we want to find the magnitude of the force Ftendon, around which point should we sum the torques?
Red Kangaroos are quite large, with masses of around 80 kg. They also have very extended feet, around 30 cm in length. Consider the forces which act on such feet when the kangaroo stands at rest. We will ignore the weight of the feet themselves, because they are small compared to the loads they support. What is the magnitude of the force applied by the tendon at the back end of the foot?
Red Kangaroos are quite large, with masses of around 80 kg. They also have very extended feet, around 30 cm in length. Consider the forces which act on such feet when the kangaroo stands at rest. We will ignore the weight of the feet themselves, because they are small compared to the loads they support. If the kangaroo is going to accelerate upward, it must make the normal force larger than its weight. If this kangaroo wants to accelerate upward with an acceleration of 5 m/s2 (rather than just stand at rest), how large must the normal force acting on each leg be?

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