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Concepts Of Physics MCQ Edition [Volume 1]PhysicsCentre of Mass, Linear momentum, Collision

For which of the following situations is the centre of mass of a rod definitively not located at its geometric centre?

Options

  1. AThe density increases continuously from left to right.
  2. BThe density increases from left to right up to the centre, and then decreases.
  3. CThe density decreases continuously from left to right.
  4. DThe density decreases from left to right up to the centre, and then increases.

Correct answer

C. The density decreases continuously from left to right.

Step-by-step solution

Let the rod be placed along the x-axis from x = -L/2 to x = L/2 . The geometric centre is at x = 0 . The position of the centre of mass is given by x_ cm = _ -L/2 ^ L/2 x (x) dx _ -L/2 ^ L/2 (x) dx . If the density (x) increases continuously from left to right, then (x) > (-x) for all x > 0 . The numerator of x_ cm can be written as ₀^ L/2 x ( (x) - (-x)) dx . Since (x) > (-x) for x > 0 , this integral is strictly positive, meaning x_ cm > 0 . Thus, the centre of mass is definitively to the right of the geometric c

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