Question

A child of mass \(M\) is swinging on a swing of length \(L\) to a maximum deflection angle of \(\theta .\) A man of mass \(4 M\) is swinging on a similar swing of length \(L\) to a maximum angle of \(2 \theta .\) Each swing can be treated as a simple pendulum undergoing simple harmonic motion. If the period for the child's motion is \(T\), then the period for the man's motion is a) \(T\). b) \(2 T\). c) \(T / 2\). d) \(T / 4\).

Short answer

(Ignore any air resistance.) a) T b) 2T c) T/2 d) 4T Answer: (a) T

Step-by-step solution

Write the formula for the period of a simple pendulum

For a simple pendulum undergoing simple harmonic motion, the formula for the period is given by: T = 2π√(L/g) where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity (approximately 9.81 m/s²).

Identify the known variables for the child and man pendulums

For the child's pendulum, we have: Mass: M Length: L Period: T For the man's pendulum, we have: Mass: 4M (4 times the child's mass) Length: L (same length as the child's pendulum) Period: Unknown, we'll call it T'

Observe that the pendulum's mass is not in the formula

Since the formula for the period of a simple pendulum does not include the mass (T = 2π√(L/g)), we don't have to worry about the difference in mass between the child and the man. Both pendulums have the same length, L, so their periods should be the same.

Calculate the period of the man's pendulum

Since the man's pendulum has the same length as the child's pendulum and the mass is not in the formula, we can conclude that the period of the man's pendulum is the same as the period of the child's pendulum: T' = T Therefore, the correct answer is (a) T.

Key concepts

Pendulum Period

The period of a pendulum is a fundamental concept in the study of simple harmonic motion. A pendulum's period is the time it takes to make one complete cycle of movement, from one extreme to the other and back again. It is often denoted by the letter 'T' and is measured in seconds.

The period of a pendulum is very important because it is a constant value for a given pendulum and can be used to understand the motion of objects that move in a repetitive pattern. The period is independent of certain variables, such as the mass of the pendulum bob and the amplitude of the swing—provided the amplitude is small. This constancy makes pendulums useful for timekeeping and scientific measurements.

Importance in a Real-World ScenarioIn a practical situation, such as with a child and a man swinging on swings, the constant nature of the period for a given length is why both their swings have the same period, regardless of their different masses. This can be counterintuitive at first glance because one might expect a heavier object to move more slowly, but in the case of the pendulum's period, this is not a factor.

Simple Pendulum Formula

The formula for the period of a simple pendulum is a simple yet powerful expression in physics. It is given by the relation:\[ T = 2\pi\sqrt{\frac{L}{g}} \]

In this formula, 'T' represents the period of the pendulum, 'L' is the length of the pendulum string, and 'g' is the acceleration due to gravity. What's important to note here is that the formula shows a direct relationship between the period and the length of the pendulum: the longer the pendulum, the longer the period. Conversely, the period is inversely related to the acceleration due to gravity. A higher 'g' value, as you might find on a planet more massive than Earth, would lead to a shorter period.

The simplicity of this formula is key to understanding pendulum motion. It encapsulates the essence of simple harmonic motion in a way that can be practically applied, such as determining the length required for a clock pendulum to achieve a specific time interval per swing.

Acceleration Due to Gravity

Acceleration due to gravity, denoted by 'g', is a crucial factor in the study of physics, especially when examining motion under gravity's influence. On the surface of the Earth, 'g' is approximately 9.81 meters per second squared (m/s²), and it represents the acceleration of an object solely under the influence of Earth's gravity.

This acceleration is what pulls the pendulum back towards its lowest point when it swings. It is a constant force that acts on every object with mass, and in the context of simple harmonic motion, it provides the restoring force that creates the oscillation.

Why 'g' MattersUnderstanding 'g' is vital because it not only affects pendulum motion but all aspects of an object's free fall or projectile motion. For example, the exact value of 'g' can differ depending on altitude and latitude, which can lead to slight variations in the period of pendulums at different locations on Earth. Despite such variations, when conducting pendulum experiments or creating precise timekeeping instruments, the value of 'g' must be taken into account to ensure accurate results.