Suppose a pendulum is pulled to one side and released at \(t_1\text{.}\) At \(t_2\text{,}\) the pendulum has swung halfway back to a vertical position. At \(t_3\text{,}\) the pendulum has swung all the way back to a vertical position. Rank the three instants in time by the magnitude of the centripetal acceleration, from greatest to least.
You swing a bucket of water with total mass m in a vertical circle with a radius given by the length of your arm, \(L\text{.}\) Determine the slowest speed at which you can swing the bucket without water falling on you. For that speed, determine the tension in your arm \(T\) at the bottom of the swing.
A conical pendulum is formed by attaching a ball of mass \(m\) to a string of length \(L\text{,}\) then allowing the ball to move in a horizontal circle of radius \(r\text{.}\)
A car starts from rest on a curve with a radius of \(120 \mathrm{~m}\) and tangential acceleration of \(1.5 \mathrm{~m/s^2}\text{.}\) Through what angle will the car have traveled when the magnitude of its total acceleration is \(3 \mathrm{~m/s^2}\text{?}\)
Represent the situation physically. The figure below shows the tangential and centripetal acceleration for the final position of the car, when the magnitude of the acceleration is \(3 \mathrm{~m/s^2}\text{.}\) At this instant, the centripetal acceleration should be larger than the tangential acceleration in order for the two components to add together properly.
Solve unknown(s) symbolically. The tangential acceleration is known and constant, so the centripetal acceleration at the end is the thing you want to know.
Is your numerical answer reasonable? This is just under one radian, so just under one sixth of a full circle. At first glance, this might seem small, but the car is moving around a pretty big circle, and will in fact have traveled just over 100 m by the time it finishes this acceleration, which seems like a reasonable distance for a car to move at everyday speeds and accelerations.
There are a lot of interesting possibilities for making sense of this symbolic answer, but the one that stands out is the numerator, which is a difference between the squares of the accelerations. In particular, the special case where \(a = a_t\) makes the numerator vanish, which means \(\Delta\theta = 0\text{.}\) This makes sense from a physical standpoint because \(a = a_t\) corresponds to the acceleration being entirely tangential, which is only true when the velocity is zero, before the car starts moving, when it will not have traveled any (angular) distance at all!
The crankshaft (\(r = 3 \mathrm{~cm}\)) in a race car accelerates uniformly from rest to \(3000 \mathrm{~rpm}\) in \(2.0 \mathrm{~s}\text{.}\) How many revolutions does it make?
The Brazilian three-banded armadillo has a hard outer exterior, and when it senses danger, it rolls into a ball for protection. Suppose that Army the armadillo rolls into a ball (with a \(5 \mathrm{~cm}\) radius) while on the slope of a hill. As a result, they begin to roll down the hill; and after \(15 \mathrm{~s}\text{,}\) they have undergone \(45.8\) revolutions. What is Army’s angular speed at this time?
