P - U2

In the left figure, a ring is attached to a board of uniform density. The ring is then moved and reattached so it is centered on the board, as shown in the right figure. The ring and the board have non-negligible masses. Which of the following could be a possible value of the change in the location of the center of mass of the ring-board system between the left and right figures?

• Zero

• 10 cm

• 20 cm

• 30 cm

Answer explanation

Correct. In the left figure, the center of mass of the ring-board system must be between the center of mass of the ring (at 30cm) and the center of mass of the board (at 50cm). In the right figure, the centers of mass of the board and the ring are both at 50cm, so the center of mass of the system is at 50cm. Therefore, the change in the center of mass must be less than 20cm but greater than zero.

Two spheres, with masses indicated in the figure, are initially at the positions on the x-axis shown in the top figure. Each sphere is then moved a distance Δx toward the other, as shown in the bottom figure. Which of the following correctly indicates how the location of the center of mass of the two-sphere system changes between the top figure and the bottom figure?

• Moves to the left by an amount less than delta x

• Moves to the left by an amount greater than delta x

• Moves to the right by an amount less than delta x

• Moves to the right by an amount greater than delta x

Answer explanation

Correct. Because both spheres move the same distance, the sphere with the larger mass will have a greater effect on the center of mass. Therefore, the center of mass will move to the left because the more massive sphere moves to the left. The center of mass changes by an amount less than Δx because the spheres move in opposite directions.

The centers of a 2.0kg sphere and a 10kg are separated by 2.0m, as shown in the figure. How far from the center of the 2.0kg sphere is the center of mass of the two-sphere system located?

• 0.3 m

• 1.7 m

• 1 m

• 2 m

Answer explanation

Correct. Using the centers of the spheres as the positions of their masses and substituting values into the formula for the center of mass yields:

A bucket rests on top of a crate, which rests on a floor, as shown in the diagram. The bucket is lifted off the crate. How does the force due to microscopic interactions between the crate and the floor change, if at all, as the bucket is lifted off the crate?

• It decreases.

• It increases.

• It stays the same.

• It could either increase or decrease depending on what the items are made of.

Answer explanation

Correct. The force due to microscopic interactions is the normal force between the crate and the floor. When the bucket is removed, the floor no longer needs to support its weight, and the magnitude of the normal force decreases.

The left figure shows a block at rest on a rough inclined surface. The block is attached to a wall by a string of negligible mass. The vector sum of the forces is zero. The right figure shows the free-body diagram for the block. The string is cut and the vector sum of the forces on the block remains zero. Which of the following free-body diagrams could show the forces exerted on the block after the string is cut?

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Answer explanation

Correct. This diagram shows forces whose vector sum is equal to zero. The horizontal components of the diagonal forces cancel each other, and the vertical components of the diagonal forces cancel the downward force.

A platform of mass m is held at rest by two ropes connected to its ends, as shown in the diagram on the left. A block, also of mass m, is placed on the center of the platform. A student draws a free-body diagram representing the forces exerted on the platform while it is at rest, as shown in the diagram on the right. Which of the following claims best explains why the diagram is incorrect?

• The tension forces from the ropes do not point in the correct directions.

• The diagram does not include components of the tension forces in the ropes.

• The horizontal forces are unbalanced.

• The vertical forces are unbalanced.

Answer explanation

Correct. The total of the upward vertical components is greater than the total of the downward vertical components (4 units on the grid upward and 2 units downward). As drawn, the platform would have an upward acceleration, but the platform is at rest.

An astronaut jumps from the surface of the Moon and follows the path shown by the dashed line. At Point P, the astronaut is at the highest point in the path. Air resistance is negligible. Is the astronaut in equilibrium? Why or why not?

• 
  Yes, because the astronaut only has a constant horizontal velocity.

• Yes, because the astronaut’s vertical velocity is instantaneously zero.

• No, because the astronaut has a component of velocity in the horizontal direction.

• 
  No, because there is a downward net force on the astronaut.

Answer explanation

Correct. The downward force of gravity is always present so the astronaut is not in equilibrium in the vertical direction at any point in its motion.

A box moves with constant speed to the right along a horizontal surface with negligible friction. Two forces, with magnitudes F1 and F2, are exerted on the box at angles θ1 and θ2, respectively, as shown. Which of the following is a correct expression for F1 in terms of F2?

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Answer explanation

Correct. The box is moving with constant speed, which means zero acceleration. Therefore, the net force must be zero and the horizontal components of the applied forces must be equal in magnitude. The horizontal component of F1 is F1cosθ1 and the horizontal component of F2 is F2sinθ2. Setting them equal yields:

A spacecraft in deep space fires its thruster, which ejects fuel from the spacecraft. The ejected fuel exerts a constant force on the spacecraft and the total mass of the spacecraft decreases at a constant rate. Which of the following graphs best represents the acceleration of the spacecraft as a function of time while the thruster is firing?

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Answer explanation

Correct. The acceleration is proportional to the net force, which is constant, and inversely proportional to the mass. Because the mass is decreasing with time at a constant rate, the acceleration will be increasing.

A person exerts a force Fp on a rope to pull a block of mass m up a ramp. The rope makes an angle θ2 to the ramp and the ramp makes an angle θ1 to the horizontal, as shown. The force of kinetic friction between the block and the ramp has magnitude Ff. Which of the following expressions is equal to the magnitude of the acceleration a of the block?

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Answer explanation

Correct. A component of the student’s pull is directed up the ramp, while the force of friction and a component of gravity are directed down the ramp. Taking the positive x-direction to be up the ramp and substituting forces into Newton’s second law yields: