In the following physical systems that undergo simple harmonic motion, state which physical quantity or quantities
you think will influence the period of motion and in what manner. Briefly justify your predictions using physical arguments rather than equations.
a. mass-spring system (oscillating horizontally on a frictionless surface)
b. simple pendulum
The simple harmonic motion is by definition the motion of a point which is subjected to a force that is proportional to the deviation of the point from the equilibrium position``.
For mass spring system the force is
`F = -kx rArr m*a +kx =0 rArr m*(d^2x)/(dt^2) +k*x =0`
`(d^2x)/(dt^2) + (k/m)*x =0` or ` ` `(d^2x)/(dt^2) +omega^2*x =0`
`T = (2*pi)/omega =(2*pi)*sqrt(m/k)` (1)
For simple pendulum the force is
`F = -mg*sin(theta)= -m*g*theta` (for small angles)
`m*a + m*g*theta =0 rArr m*l*epsilon +m*g*theta =0`
` m*l*(d^2theta)/(dt^2) +m*g*theta =0`
`(d^2theta)/dt^2 +(g/l)*theta =0 => (d^2theta)/(dt^2) +omega^2*theta =0`
`T = (2*pi)/omega = 2*pi*sqrt(l/g)` (2)
For the spring-mass system the period (1) is influenced by the values of the mass and of the spring constant. Increasing the mass increases the period, while increasing the spring constant decreases the period.
For the simple pendulum system the period (2) is influenced by the values of the length of the pendulum and of the gravitational acceleration. Increasing the length of the pendulum increases the period, while increasing the gravitational acceleration decreases the period.