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Chapter 5: Problem 39

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.

Short Answer

Expert verified
Answer: The tension in the string at the bottom of the swing is found using the formula \(T = m(a_c+g)\), where \(m\) is the mass of the bob, \(a_c\) is the centripetal acceleration, and \(g\) is the gravitational constant. Using the given values, we can find the tension force \(T\). The tension is greater than the weight of the bob because, at the bottom of the pendulum's swing, it has to counterbalance both the gravitational force and provide the necessary centripetal force for the bob to continue in its circular motion.

Step by step solution

01

Calculate Centripetal Acceleration

We have the speed \(v\) (1.6 m/s) and radius \(r\) (0.80 m) of the pendulum. Using the formula for centripetal acceleration, we get \(a_c = \dfrac{v^2}{r} = \dfrac{(1.6 \,\text{m/s})^2}{0.80 \,\text{m}}\).
02

Apply Newton's Second Law

Newton's second law states that \(F_{net} = ma\), where \(F_{net}\) is the net force acting on the bob, \(m\) is the mass, and \(a\) is the centripetal acceleration. At the bottom of the swing, the net force will be the difference between the tension force (\(T\)) and the gravitational force (\(mg\)), thus \(F_{net} = T - mg\).
03

Solve for Tension Force

We substitute our known values into Newton's second law equation and solve for \(T\): \(ma_c = T-mg \implies T = m(a_c+g)\) Using the given bob's mass \(m\) (1.0 kg), the gravitational constant \(g\) (9.81 m/s²) and the calculated centripetal acceleration \(a_c\), we can find the tension force \(T\).
04

Explain why tension is greater than the weight of the bob

At the bottom of the pendulum's swing, the tension in the string has to not only counterbalance the gravitational force acting on the bob but also provide the necessary centripetal force for circular motion. This additional force is required to keep the bob moving in a circular path. Consequently, the tension in the string is greater than the weight of the bob at the bottom of the swing.

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Most popular questions from this chapter

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