Questions: An object travels in a circle of radius r at constant speed v. a. By what factor does the object's acceleration change if its speed is doubled and the radius is unchanged? b. By what factor does the acceleration change if the radius of the circle is doubled and its speed is unchanged?

Solution Steps

The centripetal acceleration \(a\) of an object moving in a circle of radius \(r\) at a constant speed \(v\) is given by: \[ a = \frac{v^2}{r} \]

If the speed \(v\) is doubled, the new speed becomes \(2v\). Substituting \(2v\) into the formula for centripetal acceleration: \[ a' = \frac{(2v)^2}{r} = \frac{4v^2}{r} \] The new acceleration \(a'\) is four times the original acceleration \(a\): \[ a' = 4a \] Thus, the factor by which the acceleration changes is 4.

If the radius \(r\) is doubled, the new radius becomes \(2r\). Substituting \(2r\) into the formula for centripetal acceleration: \[ a'' = \frac{v^2}{2r} = \frac{v^2}{2r} \] The new acceleration \(a''\) is half the original acceleration \(a\): \[ a'' = \frac{a}{2} \] Thus, the factor by which the acceleration changes is \(\frac{1}{2}\).

Final Answer