Question

A small sphere with mass \(m\) is attached to a massless rod of length \(L\) that is pivoted at the top, forming a simple pendulum. The pendulum is pulled to one side so that the rod is at an angle \(\theta\) from the vertical, and released from rest. (a) In a diagram, show the pendulum just after it is released. Draw vectors representing the \(forces\) acting on the small sphere and the \(acceleration\) of the sphere. Accuracy counts! At this point, what is the linear acceleration of the sphere? (b) Repeat part (a) for the instant when the pendulum rod is at an angle \(\theta\)/2 from the vertical. (c) Repeat part (a) for the instant when the pendulum rod is vertical. At this point, what is the linear speed of the sphere?

Short answer

The linear acceleration just after release is \( g \sin \theta \) and at \( \theta/2 \) is \( g \sin(\theta/2) \). The linear speed when vertical is \( \sqrt{2gL(1-\cos \theta)} \).

Step-by-step solution

Analyze the forces and acceleration when released

When the pendulum is released, it is at the angle \( \theta \). The forces acting on the sphere are the gravitational force \( mg \) acting downward and the tension \( T \) in the rod acting along the rod. The acceleration \( a \) can be decomposed into tangential and radial components. Just after release, the tangential component of acceleration is given by \( a = g \sin \theta \). Since it starts from rest, \( v = 0 \) initially.

Analyze the forces and acceleration at \( \theta/2 \)

When the pendulum is at \( \theta/2 \) from the vertical, the forces still include the gravitational force \( mg \) downward and tension \( T \) along the rod. The tangential acceleration is reduced to \( a = g \sin(\theta/2) \). The sphere has gained speed by now, but we continue focusing only on forces for this part. The radial acceleration is related to the speed but is not needed here.

Analyze the forces and speed when vertical

When the pendulum is vertical, the forces include gravity \( mg \) downward and the force due to tension \( T \) which is maximal here. At this point, since the entire gravitational potential energy is now kinetic, we find the linear speed \( v \) using energy conservation: \( mgh = \frac{1}{2} mv^2 \). Solving gives \( v = \sqrt{2gL(1-\cos \theta)} \), where \( h = L(1-\cos \theta) \) is the vertical height the sphere fell.

Key concepts

Forces on Pendulum

In a simple pendulum, the forces acting on the small sphere are critical to understanding its motion. There are two primary forces at play: gravity and tension.
Gravity pulls the sphere downward with a force equal to its weight, given by \(mg\), where \(m\) is the mass of the sphere and \(g\) is the acceleration due to gravity.

• The gravitational force always acts downward toward the center of the Earth.
• Tension in the rod is directed along the rod itself, acting to pull the sphere toward the pivot point.

When the pendulum is released from an angle \(\theta\), these forces create a resultant force that affects its acceleration. Understanding these forces helps in analyzing how they contribute to pendulum motion through various angles.

Pendulum Motion

The motion of a pendulum is fascinating and involves swings or oscillations. When released from an angle \(\theta\), it follows a curved path under the influence of gravity and tension.
The pendulum starts at rest with all its potential energy at the maximum height. As it swings downwards, it converts potential energy into kinetic energy, speeding up as it approaches the lowest point.

• The motion is an example of periodic motion, repeating itself in equal intervals of time.
• Upon reaching the opposite side, the pendulum starts slowing down as kinetic energy converts back to potential energy.

This oscillation continues back and forth, demonstrating the beautiful interplay between forces and motion.

Energy Conservation

Energy conservation is a fundamental principle in understanding pendulum dynamics. In a simple pendulum, the energy exchanges between potential energy (PE) and kinetic energy (KE).
At the highest points of its swing, all energy is potential, given by \(mgh\), where \(h\) is the height from which it was released. As it descends, potential energy converts into kinetic energy, increasing its speed. When the pendulum reaches the lowest point in its swing, kinetic energy is maximized.
Using energy conservation principles, we express this as:

• At the highest point: Total energy \(= mgh\)
• At the lowest point: Total energy \(= \frac{1}{2}mv^2\)

They are equal, allowing us to calculate various parameters like speed and acceleration at different points of the pendulum's path.

Pendulum Acceleration

The acceleration of a pendulum can be understood by breaking it into its components: tangential and radial. Immediately after release, the tangential acceleration is responsible for the initial motion and is given by \(a = g \sin \theta\).

• As the pendulum moves, the angle changes, affecting the tangential acceleration.
• For an angle reduced to \(\theta/2\), the tangential acceleration becomes \(a = g \sin(\theta/2)\).

At the vertical position, the pendulum's speed peaks, and while the tangential force ceases, radial (or centripetal) acceleration takes the main role, maintaining circular motion. Understanding these changes in acceleration is key to comprehending the pendulum's dynamic behavior.