question

A boat owner pulls her boat into the dock shown, where there are six bollards to which to tie the boat. She has three ropes. She can tie the boat from the boat's center (A) to any of the bollards (B through G) along the dotted arrows shown.
Suppose the owner has tied three ropes: one rope runs to A from B, another to A from D, and a final rope from A to F. The ropes are tied such that FAB=FAD
What is the magnitude of the force provided by the third rope, in terms of Î¸?

answer

2Fabcos(theta)

question

Consider the case where m1 and m2 are both nonzero, and m2>m1. Let T1 be the magnitude of the tension in the rope connected to the block of mass m1, and let T2 be the magnitude of the tension in the rope connected to the block of mass m2. Which of the following statements is true?

answer

T1 is always equal to T2

question

Now, consider the special case where the block of mass m1 is not present. Find the magnitude, T, of the tension in the rope. Try to do this without equations; instead, think about the physical consequences.

answer

T = 0

question

For the same special case (the block of mass m1 not present), what is the acceleration of the block of mass m2?
Express your answer in terms of g, and remember that an upward acceleration should be positive.

answer

-g

question

Next, consider the special case where only the block of mass m1 is present. Find the magnitude, T, of the tension in the rope.

answer

T = 0

question

For the same special case (the block of mass m2 not present) what is the acceleration of the end of the rope where the block of mass m2 would have been attached?
Express your answer in terms of g, and remember that an upward acceleration should be positive.

answer

g

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Next, consider the special case m1=m2=m. What is the magnitude of the tension in the rope connecting the two blocks?
Use the variable m in your answer instead of m1 or m2.

answer

T = mg

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For the same special case (m1=m2=m), what is the acceleration of the block of mass m2?

answer

a2 = 0

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Finally, suppose m1â†’âˆž, while m2 remains finite. What value does the the magnitude of the tension approach?

answer

2m2g

question

Block 1, of mass m1, is connected over an ideal (massless and frictionless) pulley to block 2, of mass m2, as shown. Assume that the blocks accelerate as shown with an acceleration of magnitude a and that the coefficient of kinetic friction between block 2 and the plane is Î¼.
Find the ratio of the masses m1/m2.
Express your answer in terms of some or all of the variables a, Î¼, and Î¸, as well as the magnitude of the acceleration due to gravity g.

answer

m1/m2 = a+g(sinÎ¸+Î¼cosÎ¸)/gâˆ’a

question

When two objects slide by one another, which of the following statements about the force of friction between them, is true?

answer

The frictional force is always equal to Î¼kn.

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When two objects are in contact with no relative motion, which of the following statements about the frictional force between them, is true?

answer

The frictional force is determined by other forces on the objects so it can be either equal to or less than Î¼sn.

question

When a board with a box on it is slowly tilted to larger and larger angle, common experience shows that the box will at some point "break loose" and start to accelerate down the board.
The box begins to slide once the component of gravity acting parallel to the board Fg just begins to exceeds the maximum force of static friction. Which of the following is the most general explanation for why the box accelerates down the board?

answer

The force of kinetic friction is smaller than that of maximum static friction, but Fg remains the same.

question

Consider a problem in which a car of mass M is on a road tilted at an angle Î¸. The normal force

answer

is found using âˆ‘F =Ma

question

What is the velocity of the mass at a time t? You can work this out geometrically with the help of the hints, or by differentiating the expression for r(t) given in the introduction.

answer

v (t) = âˆ’RÏ‰sin(Ï‰t)i^+RÏ‰cos(Ï‰t)j^

question

What is the velocity of the mass at a time âˆ’ t?
Express this velocity in terms of R, Ï‰, t, and the unit vectors i^ and j^.

answer

v (âˆ’t) = RÏ‰sin(Ï‰t)i^+RÏ‰cos(Ï‰t)j^

question

What is the average acceleration of the mass during the time interval from âˆ’ t to t?

answer

avg[âˆ’t,t] = âˆ’RÏ‰/t sin(Ï‰t)i^

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What is the magnitude of this acceleration in the limit of small t? In this limit, the average acceleration becomes the instantaneous acceleration.

answer

acentripÎ·l = Ï‰^2R

question

Consider the following statements:
The centripetal acceleration might better be expressed as âˆ’Ï‰2r (t) because it is a vector.
The magnitude of the centripetal acceleration is v2radial/R.
The magnitude of the centripetal acceleration is v2tangential/R.
A particle that is going along a path with local radius of curvature R at speed s experiences a centripetal acceleration âˆ’s2/R.
If you are in a car turning left, the force you feel pushing you to the right is the force that causes the centripetal acceleration.
Identify the statement or statements that are false.

answer

b and e

question

Consider the video demonstration that you just watched. Which of the following changes would make it more likely for the ball to hit both the white can and the green can?

answer

None of the above
By Newton's first law, after it has left the circular track, the ball will travel in a straight line until it is subject to a nonzero net force. Thus, the ball can only hit the white can, because that is the only can in the ball's straight-line path.

question

A load of bricks with mass m1 = 14.0 kg hangs from one end of a rope that passes over a small, frictionless pulley. A counterweight of mass m2 = 29.0 kg is suspended from the other end of the rope, as shown in the figure. (Figure 1) The system is released from rest. Use g = 9.80 m/s2 for the magnitude of the acceleration due to gravity.
What is the magnitude of the upward acceleration of the load of bricks?

answer

3.42 m/s^2

question

A load of bricks with mass m1 = 14.0 kg hangs from one end of a rope that passes over a small, frictionless pulley. A counterweight of mass m2 = 29.0 kg is suspended from the other end of the rope, as shown in the figure. (Figure 1) The system is released from rest. Use g = 9.80 m/s2 for the magnitude of the acceleration due to gravity.
What is the tension in the rope while the load is moving?

answer

185 N

question

A large crate with mass m rests on a horizontal floor. The static and kinetic coefficients of friction between the crate and the floor are Î¼s and Î¼k, respectively. A woman pushes downward on the crate at an angle Î¸ below the horizontal with a force F
What is the magnitude of the force vector F required to keep the crate moving at constant velocity?

answer

F = Î¼kmg/ (cos(Î¸)âˆ’Î¼ksin(Î¸))

question

If Î¼s is greater than some critical value, the woman cannot start the crate moving no matter how hard she pushes. Calculate this critical value of Î¼s.

answer

Î¼s = cot(Î¸)

question

Block A in the figure (Figure 1) weighs 1.22 N and block B weighs 3.69 N . The coefficient of kinetic friction between all surfaces is 0.298.
Find the magnitude of the horizontal force Fâƒ— necessary to drag block B to the left at constant speed if A rests on B and moves with it (figure (a)).

answer

F = 1.46 N

question

Find the magnitude of the horizontal force Fâƒ— necessary to drag block B to the left at constant speed if A is held at rest (figure (b)).

answer

F = 1.83 N

question

A small remote-control car with a mass of 1.64 kg moves at a constant speed of v = 12.0 m/s in a vertical circle inside a hollow metal cylinder that has a radius of 5.00 m
What is the magnitude of the normal force exerted on the car by the walls of the cylinder at point A (at the bottom of the vertical circle)?

answer

63.3 N

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What is the magnitude of the normal force exerted on the car by the walls of the cylinder at point B (at the top of the vertical circle)?

answer

31.2 N

question

A chandelier with mass m is attached to the ceiling of a large concert hall by two cables. Because the ceiling is covered with intricate architectural decorations (not indicated in the figure, which uses a humbler depiction), the workers who hung the chandelier couldn't attach the cables to the ceiling directly above the chandelier. Instead, they attached the cables to the ceiling near the walls. Cable 1 has tension T1 and makes an angle of Î¸1 with the ceiling. Cable 2 has tension T2 and makes an angle of Î¸2 with the ceiling.
Find an expression for T1, the tension in cable 1, that does not depend on T2.

answer

T1 = mgcos(Î¸2)/sin(Î¸1+Î¸2)

question

A transport plane takes off from a level landing field with two gliders in tow, one behind the other. The mass of each glider is 700 kg, and the total resistance (air drag plus friction with the runway) on each may be assumed constant and equal to 2200 N . The tension in the towrope between the transport plane and the first glider is not to exceed 12000 N.
If a speed of 40 m/s is required for takeoff, what minimum length of runway is needed?

answer

150 m

question

What is the tension in the towrope between the two gliders while they are accelerating for the takeoff?

answer

6000 N

question

You want to move a heavy box with mass 30.0 kg across a carpeted floor. You pull hard on one of the edges of the box at an angle 30âˆ˜ above the horizontal with a force of magnitude 240 N, causing the box to move horizontally. The force of friction between the moving box and the floor has magnitude 41.5 N . What is the box's acceleration just after it begins to move?
Find both the x and y components of the acceleration of the box.

answer

ax,ay = 5.54,0 m/s2, m/s2

question

A normal walking speed is around 2.0 m/s . How much time t does it take the box to reach this speed if it has the acceleration 5.54 m/s2 that you calculated above?

answer

t = 0.36 s

question

Find Fmin, the constant force with the least magnitude that must be applied to the board in order to pull the board out from under the the box (which will then fall off of the opposite end of the board)

answer

Fmin = Î¼sg(m1+m2)

question

Consider the system shown in the figure (Figure 1) . Block A weighs 39.7 N and block B weighs 33.7 N . Once block B is set into downward motion, it descends at a constant speed.
Calculate the coefficient of kinetic friction between block A and the tabletop.

answer

Î¼ = 0.849

question

A cat, also of weight 39.7 N , falls asleep on top of block A. If block B is now set into downward motion, what is its acceleration magnitude?

answer

a = 2.92 m/s2

question

A cat, also of weight 39.7 N , falls asleep on top of block A. If block B is now set into downward motion, what is its acceleration direction?

answer

upwards

question

The Cosmoclock 21 Ferris wheel in Yokohama City, Japan, has a diameter of 100 m. Its name comes from its 60 arms, each of which can function as a second hand (so that it makes one revolution every 60.0 s).
Find the speed of the passengers when the Ferris wheel is rotating at this rate.

answer

A) v = Ï‰r = (2Ï€rads / 60s) * (100m / 2) = 5.24 m/s

question

A passenger weighs 862 N at the weight-guessing booth on the ground. What is his apparent weight at the lowest point on the Ferris wheel?

answer

B) F = m(g + a) = m(g + vÂ²/r) = (862N/9.8m/sÂ²)(9.8m/sÂ² + (5.24m/s)Â² / 50m) = 910 N

question

What is his apparent weight at the highest point on the Ferris wheel?

answer

C) F = m(g - a) = 814 N

question

What would be the time for one revolution if the passenger's apparent weight at the highest point were zero?

answer

D) F = 0 when g - a = 0, or g = Ï‰Â²r
9.8 m/sÂ² = Ï‰Â² * 50m
Ï‰ = 0.443 rad/s
T = 2Ï€/Ï‰ = 2Ï€ / 0.443rad/s = 14.2 s

question

What then would be the passenger's apparent weight at the lowest point?

answer

E) If g - a = 0, then g + a = 2g and apparent weight = 2 * 862N = 1724 N

question

An adventurous archaeologist crosses between two rock cliffs by slowly going hand-over-hand along a rope stretched between the cliffs. He stops to rest at the middle of the rope (Figure 1) . The rope will break if the tension in it exceeds 3.00Ã—104 N , and our hero's mass is 90.3 kg .
If the angle between the rope and the horizontal is Î¸ = 11.6 âˆ˜, find the tension in the rope.

answer

3.00 x 10^4 = 30000 N
mg = 90.3 x 9.8 = 884.94 N
884.94/2 = 442.47
442.47/sin(11.6) = 2200N each side

question

What is the smallest value the angle Î¸ can have if the rope is not to break?

answer

arcsin (442.47/30000) = 0.845 degrees

question

If the coefficient of static friction between a table and a uniform massive rope is Î¼s, what fraction of the rope can hang over the edge of the table without the rope sliding?

answer

r = Î¼s/(1+Î¼s)

question

A road heading due east passes over a small hill. You drive a car of mass m at constant speed v over the top of the hill, where the shape of the roadway is well approximated as an arc of a circle with radius R. Sensors have been placed on the road surface there to measure the downward force that cars exert on the surface at various speeds. The table gives values of this force versus speed for your car is shown in the table below. Treat the car as a particle.
Speed (m/s) 6.00 8.00 10.0 12.0 14.0 16.0
Force (N) 8100 7690 7050 6100 5200 4200
Select a way to represent the data from figure as a straight line. You might need to raise the speed, the force, or both to some power.

answer

n vs. v2

question

Use your graph from the part B to calculate m.

answer

m = 897 kg

question

Use your graph from the part B to calculate R.

answer

R = 49.6 m

question

What maximum speed can the car have at the top of the hill and still not lose contact with the road?

answer

v = 22.0 ms

question

In the figure (Figure 1) a worker lifts a weight w by pulling down on a rope with a force Fâƒ— . The upper pulley is attached to the ceiling by a chain, and the lower pulley is attached to the weight by another chain. The weight is lifted at constant speed. Assume that the rope, pulleys, and chains all have negligible weights.
In terms of w, find the tension in the lower chain.

answer

w

question

In terms of w, find the tension in upper chain.

answer

w

question

In terms of w, find the magnitude of the force Fâƒ— if the weight is lifted at constant speed.

answer

w/2

question

Block A in (Figure 1) weighs 1.20 N , and block B weighs 3.70 N . The coefficient of kinetic friction between all surfaces is 0.305.
Block A in (Figure 1) weighs 1.20 N , and block B weighs 3.70 N . The coefficient of kinetic friction between all surfaces is 0.305.

answer

0.305. (1.20 N + 3.70 N) = 1.49 N

question

Find the magnitude of the horizontal force Fâƒ— necessary to drag block B to the left at constant speed if A is held at rest (figure (b)).

answer

(0.305*1.20 N) + 0.305(1.20+3.70) = 1.86 N

question

On the ride "Spindletop" at the amusement park Six Flags Over Texas, people stood against the inner wall of a hollow vertical cylinder with radius 2.5 m. The cylinder started to rotate, and when it reached a constant rotation rate of 0.60 rev/s, the floor on which people were standing dropped about 0.5 m. The people remained pinned against the wall.
What minimum coefficient of static friction is required if the person on the ride is not to slide downward to the new position of the floor?

answer

Î¼min = 0.28
(g*r)/(2pi*r*rotation rate)^2

question

Does your answer in part B depend on the mass of the passenger? (Note: When the ride is over, the cylinder is slowly brought to rest. As it slows down, people slide down the walls to the floor.)

answer

no