Q: A 60 kg person stands at the left end of a 10 m long, 40 kg raft floating on water. Where is the center of mass of the system (person + raft) relative to the left end of the raft?

$$x_{CM} = \frac{60 \times 0 + 40 \times 5}{60 + 40} = 2 m$

Q: A system consists of two masses, $$m_1 = 2$ kg at x_1$ = 0 m $$and $$m_2 = 3$ kg at x_2$ = 4 m. $$What is the position of the center of mass?

$$x_{CM} = \frac{2 \times 0 + 3 \times 4}{2 + 3} = 2.4 m$

Q: In a collision between two cars, car A (1000 kg) moving at 10 m/s and car B (1500 kg) at rest, they stick together after the collision. What is their final velocity?

$$v_f = \frac{1000 \times 10 + 1500 \times 0}{1000 + 1500} = 4 m/s$

Question 1

A rocket sled loaded with rocks coasts on a level track without friction. If a worker at the head of the sled throws the rocks horizontally backward off the rear, what happens to the sled?

Accepted Answer

The sled speeds up.

Question 2

Which of the following best describes the law of conservation of linear momentum?

Accepted Answer

The total momentum of a system remains constant if no external force acts on it.

Question 3

A wrench of mass $$m_w$$ and a rock of mass $$m_r$$ are connected by a rope and move along a horizontal surface. If $$x_w$$ and $$x_r$$ are their respective positions, what is the formula for the center of mass $$x_{CM} of $$the system?

Accepted Answer

$$x_{CM} = \frac{m_w x_w + m_r x_r}{m_w + m_r}$$

Question 4

A diver jumps off a platform and rotates in the air. Which statement about the diver's center of mass (CM) is correct?

Accepted Answer

The CM follows a straight path determined by gravity, regardless of the diver's rotation.

Question 5

A 60 kg person stands at the left end of a 10 m long, 40 kg raft floating on water. Where is the center of mass of the system (person + raft) relative to the left end of the raft?

Accepted Answer

$$x_{CM} = \frac{60 	imes 0 + 40 	imes 5}{60 + 40} = 2 m$

Question 6

Which of the following is true about the center of mass (CM) of the human body?

Accepted Answer

The CM can shift depending on body position and distribution of mass.

Question 7

A ball of mass $$m$$ moving at velocity $$v$$ collides elastically with a stationary ball of equal mass. What are the velocities of the two balls after the collision?

Accepted Answer

The first ball stops, and the second ball moves at $$v.$$

Question 8

Which equation represents the conservation of momentum for a two-object system in one dimension?

Accepted Answer

$$m_1$$ $$v_1$$ + $$m_2$$ $$v_2$$ = $$m_1$$ $$v_1$$' + $$m_2$$ $$v_2$$'$$

Question 9

A person throws a rock while standing on a frictionless sled. If the rock is thrown forward, what happens to the sled?

Accepted Answer

The sled moves backward.

Question 10

Which of the following best describes the motion of the center of mass of a system of particles?

Accepted Answer

It moves as if all the mass of the system were concentrated at the CM and all external forces acted at that point.

Question 11

A 70 kg gymnast hangs vertically from a bar. If her head and neck account for 7% of her body mass and are located 1.5 m above her hands, what is the contribution of her head and neck to the vertical position of her center of mass?

Accepted Answer

$$0.07 	imes 1.5 = 0.105 m$

Question 12

A system consists of two masses, $$m_1 = 2$ kg at x_1$ = 0 m $$and $$m_2 = 3$ kg at x_2$ = 4 m. $$What is the position of the center of mass?

Accepted Answer

$$x_{CM} = \frac{2 	imes 0 + 3 	imes 4}{2 + 3} = 2.4 m$

Question 13

In a collision between two cars, car A (1000 kg) moving at 10 m/s and car B (1500 kg) at rest, they stick together after the collision. What is their final velocity?

Accepted Answer

$$v_f = \frac{1000 	imes 10 + 1500 	imes 0}{1000 + 1500} = 4 m/s$

Question 14

Which of the following is NOT a requirement for the conservation of momentum in a system?

Accepted Answer

The system must have constant velocity.

Question 15

A person stands on a frictionless surface and throws a ball of mass $$m$$ with velocity $$v. If $$the person has mass $$M$$, what is the recoil velocity of the person?

Accepted Answer

$$v_{person} = -\frac{m}{M} v$$

Question 16

Which statement about the center of mass (CM) and gravity is correct?

Accepted Answer

The CM is the average position of all the mass in a body and is the point where gravity can be considered to act for translational motion.

Question 17

A system consists of three particles: $$m_1 = 1$ kg at (0$$,0)$$, m_2$ = 2 kg at$$ $$(2,0)$$, and $$m_3 = 3$ kg at (0$$,3)$$. What are the coordinates of the center of mass?

Accepted Answer

$$(x_{CM}, y_{CM}) = (\frac{1 	imes 0 + 2 	imes 2 + 3 	imes 0}{1 + 2 + 3}, \frac{1 	imes 0 + 2 	imes 0 + 3 	imes 3}{1 + 2 + 3}) = (0.67, 1.5)$$

Question 18

A 50 kg child and a 70 kg adult sit at opposite ends of a 4 m long seesaw. Where is the center of mass of the system relative to the child's position?

Accepted Answer

$$x_{CM} = \frac{50 	imes 0 + 70 	imes 4}{50 + 70} = 2.33 m$

M06Linear Momentum, Collisions, and Center of Mass

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