Chapter 5: Problem 48

Find the tangential acceleration of a freely swinging pendulum when it makes an angle \(\theta\) with the vertical.

Short Answer

Expert verified

Answer: The tangential acceleration, \(a_t\), of a freely swinging pendulum when it makes an angle θ with the vertical is given by the formula \(a_t = g \sin{\theta}\), where g is the acceleration due to gravity.

Step by step solution

Draw a diagram of the pendulum and identify the forces acting on it

Draw a pendulum of length L and mass m. When the pendulum makes an angle \(\theta\) with the vertical, the forces acting on the mass are the gravitational force (mg, acting straight downwards) and the tension in the rod (T, acting in line with the rod).

Set up the equations of motion

We can resolve these forces into two components: radial and tangential. The radial force acts inwards towards the pivot, while the tangential force acts perpendicular to the radial direction (and along the path of the mass). - The radial force component, Fr, is given by \(Fr = T - mg \cos{\theta}\). - The tangential force component, Ft, is given by \(Ft = mg \sin{\theta}\). Applying Newton's second law of motion, \(F = ma\), where 'a' is the acceleration of the mass: 1. In the radial direction: \(Fr = m a_r \Rightarrow T - mg \cos{\theta} = m \frac{v^2}{L}\). 2. In the tangential direction: \(Ft = m a_t \Rightarrow mg \sin{\theta} = m a_t\).