A ball rolls without slipping on a flat surface. If the ball's radius is doubled
Practice Questions
Q1
A ball rolls without slipping on a flat surface. If the ball's radius is doubled while keeping its mass constant, how does its moment of inertia change?
Increases by a factor of 2
Increases by a factor of 4
Increases by a factor of 8
Remains the same
Questions & Step-by-Step Solutions
A ball rolls without slipping on a flat surface. If the ball's radius is doubled while keeping its mass constant, how does its moment of inertia change?
Correct Answer: Moment of inertia increases by a factor of 4.
Step 1: Understand what moment of inertia is. It is a measure of how difficult it is to change the rotation of an object.
Step 2: Know the formula for the moment of inertia of a solid sphere, which is (2/5)MR^2, where M is mass and R is radius.
Step 3: Identify that the mass (M) of the ball remains constant in this problem.
Step 4: Recognize that if the radius (R) is doubled, we replace R with 2R in the formula.
Step 5: Substitute 2R into the moment of inertia formula: Moment of inertia = (2/5)M(2R)^2.
Step 6: Calculate (2R)^2, which equals 4R^2.
Step 7: Now the moment of inertia becomes (2/5)M(4R^2) = (8/5)MR^2.
Step 8: Compare the new moment of inertia (8/5)MR^2 with the original (2/5)MR^2. The new moment of inertia is 4 times the original moment of inertia.
Moment of Inertia – The moment of inertia is a measure of an object's resistance to changes in its rotation, depending on its mass distribution relative to the axis of rotation.
Effect of Radius on Moment of Inertia – The moment of inertia for a solid sphere is proportional to the square of its radius, meaning if the radius is doubled, the moment of inertia increases by a factor of four.
