Question

A person sit on a swing. When the person sits still, the swing oscillates back and forth at the natural frequency.A second person sit on the swing. Is the natural frequency smaller, the same or larger? To answer this question write down the physical principle, make a drawing with appropriate labeling, demonstrate that for small swing amplitudes the motion is well described by simple harmonic motion and find the mass dependence.?

Answer 1

Lets say at one situation the swing makes an angle theta from vertical so the compoents along the rope = mg cos thetta, perpendicular to the rope mg sin theta

Now mg cos theta= T , and the mg sin theta gives the nessessary restoring force

Fr= mg sin theta ( for small angles sin theta= theta in radians )

Fr= mg theta , But from circles theta= x/L , x is the arc length, L is the radius

Fr= mgx/ L This F is opposite of x (displacement from mean position)

Fr= -mgx/L

ma = -m g x/L

a= -(g/L) x

a= -w^2 x (Standard equation for SHM)

compare w= sqrt(g/L) = 2*pi*f

f = 1/2 pi ( sqrt(g/l))

So we see frequency is independent of mass, so its remains the same as the second person sits on it. Remeber the approximation of sine,only for small angles.

In the attached image its s instead of x.