Lesson 2 Objective

At the end of this section, you should be able to:

• calculate the period of simple pendulum of a given length;
• compute the period of oscillation of a spring mass system on smooth
  horizontal surface;
• define Hooke’s law;
• practically measure the periods of simple pendulumof a given length and
  spring mass systemin a laboratory or class;
• calculate the value of the acceleration due to gravity in your locality.

Brainstorming Question

Find a string or rope in your locality. Stretch the string or rope by fixing one of its ends with a fixed pole and holding the other end with your hand. We say that the string is in equilibrium. Now, flick the string either up or down. What do you observe?

key terms and concepts

• The period of a simple pendulum depends only on its length and the acceleration due to gravity. It does not depend on the mass of the bob.
• The force exerted by the spring is a restoring force. No matter which way the object is displaced from equilibrium, the spring force always acts to return the object to equilibrium.
  

String and Pulse

When you flick a string, a single disturbance called a pulse travels along the string. The string’s particles move up and down at right angles to the pulse’s horizontal motion. A wave is a series of such pulses moving through a medium.

Figure 6.2 Pulsed string wave.

Simulation

https://javalab.org/en/spring_pendulum_en/

Periodic Motion

Periodic motion is a repetitive movement, like a swinging pendulum or a ticking clock. The time for one complete cycle of motion is called the period. A simple pendulum’s period depends on its length and gravity. The period T of a simple pendulum is given by:

​​ T = $2\pi \sqrt{\frac{L}{g}}$ where:

• T = period of the pendulum
• L = length of the pendulum
• g = acceleration due to gravity

Example: What is the period of a simple pendulum with a length of 1 meter? Assume g=9.8 m/s².

Solution: Using the formula:  T = $2\pi \sqrt{\frac{1}{9.8}}$ ​​≈2π×0.32≈2.01seconds

Figure 6.3 Schematics of simple pendulum.

Simple Spring and Forces

When a mass attached to a spring is displaced from its equilibrium position, it experiences a restoring force described by Hooke’s Law:

F_rest = -kx where:

Figure 6.4 Demonstration of Hooke’s law using displacement of a spring

• F_rest= restoring force
• k = spring constant(stiffness of the spring)
• x= displacement from equilibrium

The period T of oscillation for a spring-mass system is given by: T= $2\pi \sqrt{\frac{m}{k}}$​​ where:

• T = period of the spring
• m = mass attached to the spring
• k = spring constant

Example: Calculate the period of a block with mass 0.5 kg attached to a spring with a spring constant of 100 N/m.

• Solution: Using the formula:T= $2\pi \sqrt{\frac{0.5}{100}}$ ​​≈2π×0.07≈0.44seconds