To make a pendulum, a 300 g ball is attached to one end of a string that has a length of 1.4 m

step 1 Here, for Part A, we're going to use the conservation of energy from equation 817. We can say th

To make a pendulum, a 300 g ball is attached to one end of...

To make a pendulum, a $300 \mathrm{~g}$ ball is attached to one end of a string that has a length of $1.4 \mathrm{~m}$ and negligible mass. (The other end of the string is fixed.) The ball is pulled to one side until the string makes an angle of $30.0^{\circ}$ with the vertical; then (with the string taut) the ball is released from rest. Find (a) the speed of the ball when the string makes an angle of $20.0^{\circ}$ with the vertical and (b) the maximum speed of the ball. (c) What is the angle between the string and the vertical when the speed of the ball is one-third its maximum value?

Key Concepts

Conservation of Energy

In many physics problems involving pendulums, the principle of conservation of energy is key. This principle states that in the absence of non-conservative forces (like friction or air resistance), the total mechanical energy—comprising gravitational potential energy and kinetic energy—remains constant throughout the motion. It allows us to equate the loss in gravitational potential energy due to a change in height to the gain in kinetic energy as the pendulum swings downward, and vice versa.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy due to an object's position in a gravitational field, typically defined as U = mgh where h is the vertical height relative to a chosen reference level. In pendulum problems, the change in height is determined by the length of the pendulum and the cosine of the angle it makes with the vertical, making trigonometric relationships essential for calculating the change in GPE as the pendulum swings.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object, given by the formula KE = ½mv². In the context of a pendulum, when the pendulum loses gravitational potential energy as it swings down, this energy is converted into kinetic energy, increasing the speed of the pendulum. This conversion is central to solving for the speed of the pendulum at different points in its swing.

Trigonometric Relationships in Pendulum Motion

Trigonometry plays a pivotal role in analyzing pendulum motion because the change in the vertical height of the pendulum bob depends on the cosine of the angle formed with the vertical. By relating the length of the pendulum and the cosine of the displacement angle, one can calculate the vertical distance through which the bob has fallen or risen, which is required to compute changes in gravitational potential energy.

Pendulum Dynamics

Pendulum dynamics involves understanding the motion and behavior of a pendulum under the influence of gravity. Key aspects include identifying points of maximum and minimum speeds—such as the bottom and the endpoints of the swing—and how energy is partitioned during the motion. Analyzing these aspects helps in determining parameters like the speed at any angle and the maximum speed, using energy methods and trigonometric relationships.

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