Two identical small spheres

PHYSICS · Multi-part · A Level · Physics · Mechanics and materials · Mechanics and materials

Problem

Two identical small spheres, each of mass

m

, are fixed at the points

(0,0)

and

(d,0)

in the

xy

-plane. A third sphere of mass

3m

is initially fixed at the point

(\tfrac{d}{2},\, h)

, directly above the midpoint of the line joining the first two spheres. The third sphere is then released from its fixture and allowed to slide without friction straight down the

y

-axis toward the

x

-axis. At any instant, its

y

-coordinate is

y

(with

h \ge y \ge 0

). Assume the other two spheres remain fixed. (1) Derive an expression for the

x

-coordinate of the center of mass of the three-sphere system as a function of the variable

y

. Express your result in terms of

m,\,d,\,

and

y

. (2) Show that the

x

-coordinate of the center of mass remains constant while the third sphere moves. State the numerical value of this constant in terms of

d

. (3) Derive an expression for the

y

-coordinate of the center of mass as a function of

y

and describe the path traced by the center of mass in the

xy

-plane as the third sphere descends. (4) When the third sphere reaches the

x

-axis (

y=0

), determine the final coordinates of the center of mass and explain physically why the horizontal position remained unchanged throughout the motion.

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