Experiment 4 find the value of G using a pendulum using Phet

OBJECTIVES

To determine the value of acceleration due to gravity by using a simple pendulum.

EQUIPMENT

PhET Pendulum Lab: https://phet.colorado.edu/en/simulation/pendulum-lab

You can also get to the simulations by entering in your browser: Phet, then select Physics. Then select Pendulum-lab simulation.

THEORY

Whenever a restoring force is created due to displacement of a body from its mean (equilibrium) position, the restoring force tries to bring the body back to the original (mean) position. If the restoring force is proportional to the displacement from the mean position, the resulting vibratory motion is called simple harmonic motion. A linear restoring force may be ex­pressed mathematically by the equation

F = -k Δx (1)

where F is the force (N), Δx (m) is the displacement from the equilibrium point, and k (N/m) is a proportionality con­stant. The minus sign expresses the restoring nature of the force by indicating that the direction of F is always opposite to the displacement Δx from equilibrium.

For vibratory motion we have the following definitions:

Amplitude, A: The maximum displacement from the equilibrium point on either side. (unit: m)

Cycle: One complete vibration. (no units)

Time Period, T: The time required for one complete cycle. (unit: s)

Frequency, f: The number of cycles completed per sec­ond (= 1/T). (unit: s^-1)

Further analysis of the vibrations resulting from a linear restoring force shows that the Time Pe­riod is given by:

(2 )

where T is the Time Period, m is the mass of the vibrating body, and k is the proportionality constant from Equation 1.

Two physical examples of this type of force are con­sidered in the present experiment, the simple pendulum and the spring-mass system.

SIMPLE PENDULUM

This consists of a concentrated mass sus­pended at the end of a cord of negligible weight, such as a small metal sphere on a long, thin thread, as shown in Fig. 1. The sphere has been pulled aside to a horizontal distance x, and the thread consequently makes an angle θ with the ver­tical. The vertical line through the suspension point is at X = 0, which is the equilibrium position. From Fig 1:

For small θ, we have Sin θ ≈ θ, and m g Sinθ ≈ m g θ.

Now θ = S / L, where S is the distance that the mass has moved along the arc, between the equilibrium position and the new position. For small θ, X ≈ S, and we get: θ = X / L. The net force then becomes:

m g Sinθ ≈ m g θ = m g X / L = (m g / L ) X.

In eqn. (1), the k becomes the same as mg/ L, and the equation for the time period becomes:

(3)

By measuring the value of the Time Period of a pendulum of length L, we can calculate the value of acceleration due to gravity, g.

Figure 1: Simple Pendulum