Consider a system of two blocks that have masses m 1 and m 2. Assume that the blocks are point-like particles and are located along the x axis at the coordinates x 1 and x 2 as shown (Figure 1) . In this problem, the blocks can only move along the x axis.

Part A

Find the x coordinate x cm of the center of mass of the system.

Express your answer in terms of m 1, m 2, x 1, and x 2.

x cm =

m 1 x 1+ m 2 x 2 m 1+ m 2

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Part B

If m 2≫ m 1, then the center of mass is located:

	to the left of m 1 at a distance much greater than x 2− x 1
	to the left of m 1 at a distance much less than x 2− x 1
	to the right of m 1 at a distance much less than x 2− x 1
	to the right of m 2 at a distance much greater than x 2− x 1
	to the right of m 2 at a distance much less than x 2− x 1
	to the left of m 2 at a distance much less than x 2− x 1

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Part C

If m 2= m 1, then the center of mass is located:

	at m 1
	at m 2
	half-way between m 1 and m 2
	the answer depends on x 1 and x 2

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Part D

Recall that the blocks can only move along the x axis. The x components of their velocities at a certain moment are v 1x and v 2x. Find the x component of the velocity of the center of mass ( v cm) x at that moment. Keep in mind that, in general: vx = dx / dt .

Express your answer in terms of m 1, m 2, v 1x, and v 2x.

( v cm) x =

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Part E

Suppose that v 1x and v 2x have equal magnitudes. Also, v 1→ is directed to the right and v 2→ is directed to the left. The velocity of the center of mass is then:

	directed to the left
	directed to the right
	zero
	the answer depends on the ratio m 1 m 2

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Part F

Assume that the x components of the blocks' momenta at a certain moment are p 1x and p 2x. Find the x component of the velocity of the center of mass ( v cm) x at that moment.

Express your answer in terms of m 1, m 2, p 1x, and p 2x.

( v cm) x =

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Part G

Suppose that v ⃗ cm=0. Which of the following must be true?

	| p 1 x |=| p 2 x |
	| v 1 x |=| v 2 x |
	m 1= m 2
	none of the above

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Part H

Assume that the blocks are accelerating, and the x components of their accelerations at a certain moment are a 1xand a 2x. Find the x component of the acceleration of the center of mass ( a cm) x at that moment. Keep in mind that, in general, ax = dvx / dt .

Express your answer in terms of m 1, m 2, a 1x, and a 2x.

( a cm) x =

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We will now consider the effect of external and internal forces on the acceleration of the center of mass.

Part I

Consider the same system of two blocks. An external force F ⃗ is now acting on block m 1. No forces are applied to block m 2 as shown (Figure 2) . Find the x component of the acceleration of the center of mass ( a cm) x of the system.

Express your answer in terms of the x component Fx of the force, m 1,and m 2.

( a cm) x =

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Part J

Consider the same system of two blocks. Now, there are two forces involved. An external force F 1→ is acting on block m 1 and another external force F 2→ is acting on block m 2. (Figure 3) Find the x component of the acceleration of the center of mass ( a cm) x of the system.

Express your answer in terms of the x components F 1x and F 2x of the forces, m 1 and m 2.

( a cm) x =

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Part K

Consider the previous situation. Under what condition would the acceleration of the center of mass be zero? Keep in mind that F 1x and F 2x represent the components , of the corresponding forces.

Consider the previous situation. Under what condition would the acceleration of the center of mass be zero? Keep in mind that and represent the components , of the corresponding forces.

	F 1 x =− F 2 x
	F 1 x = F 2 x
	m 1= m 2
	m 1≪ m 2

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Part L

Consider the same system of two blocks. Now, there are two internal forces involved. An internal force F ⃗ 12 is applied to block m 1 by block m 2 and another internal force F ⃗ 21 is applied to block m 2 by block m 1 (Figure 4) . Find the x component of the acceleration of the center of mass ( a cm) x of the system.

Express your answer in terms of the x components F 12x and F 21x of the forces, m 1 and m 2.

( a cm) x =