SHM question

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In an experiment I am changing the mass on a pendulum and recording the time period for one oscillation and the velocity at equilibrium posistion.

What is the theory behind mass affecting time period and velocity?

my results dont show much- dont think theyre vary accurate
time period doesnt change
velocity decreases with mass?!
What equations are involved in the theory?

Reply 1

James

Remember that the simple pendulum does not obey SHM. Its motion only approximates to SHM with small-amplitude oscillations, i.e. when the angle remains small.

If you are going to investigate theory, you need to draw a diagram, mark on forces and write down the equations of motion. You will then be able to see for yourself the impacts of mass etc.

Reply 2

Theorist

Mass doesn't affect the time period or velocity of the pendulum, as it accelerates due to gravity, which is constant; the acceleration being given by a = g sin theta.

If you look at it from a mechanical energy perspective, the potential energy of the pendulum at the point of maximum displacement is equal to its maximum kinetic energy at the mean/centre position.

GPE = KE
mgh = 0.5mv²

However, the masses cancel, to give v = sqrt(2gh).

The following statement assumes that the amplitude of the pendulum is small i.e. below about 15 degrees. The pendulum's time period is only affected by its length (I do not know the reason for this, I simply know that this is the case). The time period for a pendulum's oscillation is given by the relationship:

T = 2(pi)sqrt(l/g)

where T = time period, l = length of pendulum and g = acceleration due to gravity. If you want to find a value of g from this graph, plot a graph of T² against l. The gradient will be equal to 4(pi)²/g and therefore, g = 4(pi)²/m, where m is the gradient.

Reply 3

James

With a simplified model neglecting air resistance and friction, resolving in the e_theta direction (parallel to the direction of movement) gives:

angular acceleration = -g sin(theta)/l

When the angle is small, we can approximate sin(theta) by theta, and thus we have an equation of the form

theta double dot = -omega squared theta

(SHM)