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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 胃 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

01

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).
02

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: \(Ft = m a_t \Rightarrow mg \sin{\theta} = m a_t\).
03

Find the tangential acceleration

From the second equation, we can isolate the tangential acceleration, \(a_t\): \(mg \sin{\theta} = m a_t\) Divide both sides by m: \(g \sin{\theta} = a_t\) Tangential acceleration, \(a_t\), of the pendulum when it makes an angle \(\theta\) with the vertical is: \(a_t = g \sin{\theta}\)

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Most popular questions from this chapter

In an amusement park rocket ride, cars are suspended from 4.25 -m cables attached to rotating arms at a distance of \(6.00 \mathrm{m}\) from the axis of rotation. The cables swing out at an angle of \(45.0^{\circ}\) when the ride is operating. What is the angular speed of rotation?
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A pendulum is \(0.80 \mathrm{m}\) long and the bob has a mass of 1.0 kg. At the bottom of its swing, the bob's speed is \(1.6 \mathrm{m} / \mathrm{s} .\) (a) What is the tension in the string at the bottom of the swing? (b) Explain why the tension is greater than the weight of the bob.
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