Along a straight

MATHEMATICS · Multi-part · A Level · Mathematics · Forces and Newton's laws · Forces and Newton's laws

Problem

Along a straight, frictionless air track three small gliders are initially fixed in place at the positions shown below: • Glider 1: mass

2\,\text{kg}

x = 1\,\text{m}

• Glider 2: UNKNOWN mass

m

x = 4\,\text{m}

• Glider 3: mass

3\,\text{kg}

x = 7\,\text{m}

All positions are measured along the positive

x

-axis. A laboratory technician later places a fourth glider of mass

1\,\text{kg}

x = 5\,\text{m}

and gently releases all gliders to move freely. (a) The technician wants the center of mass of the four-glider system to be exactly at

x = 5\,\text{m}

after the fourth glider is added. Determine the value of the unknown mass

m

that satisfies this requirement. (b) Using the value of

m

found in part (a), calculate the

x

-coordinate of the center of mass of the original three-glider system (before the fourth glider is added). (c) Compute the magnitude and direction (left or right) of the displacement of the system’s center of mass caused solely by adding the fourth glider. (d) Provide a brief justification for why placing the additional glider at

x = 5\,\text{m}

was sufficient to shift—or not shift—the center of mass to the desired final location, even though internal forces among the gliders remain negligible.

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