Question

The timing device of Problem \(2.19\) is tapped to initiate motion. Determine the magnitude of the impulse required so that the motion of the device has an amplitude \(\Theta_{0}\).

Short answer

Answer: The magnitude of the impulse required to initiate the motion of the timing device with an amplitude \(\Theta_{0}\) is given by the expression: Impulse = \(I\sqrt{\frac{2g\Theta_{0}}{l}}\), where \(I\) is the moment of inertia of the device, \(g\) is the acceleration due to gravity, and \(l\) is the length of the pendulum.

Step-by-step solution

Set up the problem

Given a timing device, we need to calculate the impulse required to set the device in motion with an amplitude \(\Theta_{0}\). We will use the principle of impulse-momentum, which states that the impulse applied to a system equals the change in the system's linear or angular momentum. In mathematical terms, the impulse-momentum principle states: Impulse = change in momentum. For this problem, we'll work with angular momentum, so the equation turns into: Impulse = change in angular momentum.

Determine the initial and final angular momentum

Let's call the initial angular momentum of the device \(L_{1}\) and the final angular momentum \(L_{2}\). Since the device is initially at rest, we have \(L_{1} = 0\). At the final state, when the device reaches the amplitude \(\Theta_{0}\), the device's angular momentum will be maximum. Let's denote the moment of inertia of the device by \(I\) and the maximum angular velocity by \(\omega_{max}\). Hence, at the final state, we have \(L_{2} = I\omega_{max}\).

Calculate the change in angular momentum

Now that we have the initial and final angular momentum, we can calculate the change in angular momentum, which is given by the difference between the final and initial values: Change in angular momentum = \(L_{2} - L_{1} = I\omega_{max} - 0 = I\omega_{max}\).

Relate the amplitude and maximum angular velocity

To find a relationship between the amplitude \(\Theta_{0}\) and maximum angular velocity \(\omega_{max}\), we can use the following equation for the motion of a simple pendulum: \(\omega_{max} = \sqrt{\frac{2g\Theta_{0}}{l}}\) where \(g\) is the acceleration due to gravity, and \(l\) is the length of the pendulum.

Calculate the impulse

Now that we have a relationship between \(\omega_{max}\) and \(\Theta_{0}\), we can calculate the impulse required to initiate the motion of the device using the impulse-momentum principle: Impulse = Change in angular momentum = \(I\omega_{max}\) Substituting the expression of \(\omega_{max}\) from step 4: Impulse = \(I\sqrt{\frac{2g\Theta_{0}}{l}}\) Thus, the magnitude of the impulse required to initiate the motion of the device with an amplitude \(\Theta_{0}\) is given by: Impulse = \(I\sqrt{\frac{2g\Theta_{0}}{l}}\).