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Chapter 6: Problem 46

A pendulum consists of a small object hanging from the ceiling at the end of a string of negligible mass. The string has a length of 0.75 m. With the string hanging vertically, the object is given an initial velocity of 2.0 m/s parallel to the ground and swings upward on a circular arc. Eventually, the object comes to a momentary halt at a point where the string makes an angle \(\theta\) with its initial vertical orientation and then swings back downward. Find the angle \(\theta\) .

Short Answer

Expert verified
The angle \(\theta\) is approximately \(43.9^\circ\).

Step by step solution

01

Understand the Problem

This problem involves a pendulum moving in a gravitational field. We are to find the angle \(\theta\) at the highest point of its swing, where the pendulum momentarily halts after being given an initial horizontal velocity.
02

Apply Conservation of Energy

The principle of conservation of mechanical energy states that the total energy (kinetic + potential) in a closed system remains constant. Initially, the pendulum has only kinetic energy, and at the highest point in its swing, all energy will be stored as potential energy.
03

Calculate Initial Kinetic Energy

The initial kinetic energy \(KE_i\) can be calculated using the formula: \(KE_i = \frac{1}{2} mv^2\), where \(m\) is the mass of the object and \(v = 2.0\, \text{m/s}\) is the initial velocity.
04

Calculate Potential Energy at Highest Point

At the highest point, all the initial kinetic energy will have converted into gravitational potential energy \(PE_f\), which is given by \(PE_f = mgh\), where \(h\) is the height the object has risen, and \(g = 9.8\, \text{m/s}^2\) is the acceleration due to gravity.
05

Relate Height to Angle

The height \(h\) can be related to the angle \(\theta\) using the pendulum length \(L = 0.75\, \text{m}\) and the formula \(h = L - L\cos\theta\). Substitute this expression for \(h\) into the potential energy formula.
06

Set Up Energy Equation

From the conservation of energy, set the initial kinetic energy equal to the potential energy at the highest point: \(\frac{1}{2} mv^2 = mg(L - L\cos\theta)\). The mass \(m\) cancels out from both sides of the equation.
07

Solve for \(\cos\theta\)

Rearrange the equation: \(\frac{1}{2}v^2 = g(L - L\cos\theta)\). Simplify further into \(\cos\theta = 1 - \frac{v^2}{2gL}\). Substitute \(v = 2.0\, \text{m/s}\), \(g = 9.8\, \text{m/s}^2\), and \(L = 0.75\, \text{m}\) to solve for \(\theta\).
08

Calculate \(\theta\)

Find \(\cos\theta\) by substituting the numbers: \(\cos\theta = 1 - \frac{(2.0)^2}{2 \, (9.8)(0.75)}\). Calculate this to find \(\cos\theta\approx0.726\). Use the inverse cosine function to find \(\theta\): \(\theta = \cos^{-1}(0.726)\).
09

Find the Angle

Convert \(\cos^{-1}(0.726)\) using a calculator to get the angle: \(\theta \approx 43.9^\circ\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conservation of Energy
The conservation of energy is a fundamental principle in physics that states energy in a closed system remains constant. In the context of a pendulum, this concept helps explain how energy shifts between motion states. Initially, a moving pendulum has kinetic energy due to its motion. As it swings upward and slows down, this kinetic energy transforms into potential energy. At its highest point, the pendulum briefly stops, having converted all the kinetic energy into potential energy. This energy transformation allows us to calculate the pendulum's swing angle by knowing initial speeds without needing other properties like mass.
Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion. In the pendulum exercise, the kinetic energy is significant at the initial point of motion. The formula for kinetic energy is given by \( KE = \frac{1}{2} mv^2 \), where \( m \) represents mass and \( v \) is velocity. Given an initial velocity of \( 2.0 \text{ m/s} \), the pendulum starts with maximum kinetic energy. As the object swings upward, this kinetic energy reduces as it converts into potential energy. Understanding kinetic energy helps explain how the movement initiates with speed and transforms up the swing.
Potential Energy
Potential energy represents the stored energy of an object due to its height in a gravitational field. Initially, the pendulum's potential energy is minimal at its lowest point. As the pendulum moves upward, the potential energy increases, peaking when the motion halts. The potential energy at the highest point can be expressed as \( PE = mgh \), where \( h \) is the height gained. By calculating this height, we can determine the angle \( \theta \) in the pendulum's swing. This energy enables the pendulum to sway back and forth by converting said energy back to kinetic on descent.
Circular Arc Motion
Circular arc motion describes the path an object follows along a circular route, like a pendulum's swing. This motion in pendulums involves moving along a circular arc, where its path forms a sector of a circle. The length of the pendulum's string acts as the radius of this circle. In this motion, the angle at which the pendulum halts, \( \theta \), is found by correlating the height \( h \) the object rises using equation \( h = L - L \cos \theta \). This interaction of the pendulum's length and height defines the angles of swing. Understanding circular arc motion allows one to interpret how pendulums transform energy while following a curved trajectory.

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