simple harmonic motion and mass?

Question

Period (T) is independant of pendulum mass. why is this?

i was given this: (hint: think of galileo)

Answer 1

Think of the the pendulum as having a length l. If it is a some angle q from the vertical then you can write an equation that equates the force due to gravity along the pendulum arm with the tension force of the pendulum arm, and the mass of the bob times its acceleration to the component of gravity perpendicular to the arm. This looks like:

ma(t) = -m*g*l*sin(q) ---> a(t) = -g*l*sin(q)

So you can see that mass has no role in the acceleration, only the length of the pendulum matters. If you let q be small so that sin(q) ~ q, and assume (you can rigorously show this) that

a = Aw^2*sin(wt) where t is time and w =2*pi*f and f = frequency of the pendulum, then you can show that w = sqrt(g/l) so that

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

Answer 2

Gravity is a even tempered woman . She acts the same on light weights as well as heavy weights. The attraction and acceleration of a falling body to the earth for a big weighted pendulum is the same as a very small one. If ,,you want to skip the relative friction of air . The period of a pendulum consists of the time it take to go from the drop point to the apogee and back.....since distance =rate x time ...time= distance/rate
rate is fixed at gravitational attraction so distance is our main variable . If the string Length is increased then proportionally the time of fall is increased (if you increase the radius of a circle the circumference increases by Pi) soooooo....making the sting longer makes the period longer. You can put a
5 gram weight on the end or a 500gram weight on the end and the period will still be the same. I think a better hint would have been to consider the work of Newton and his work with motion (especially gravity)....Well have a good day...from the E.

addenda: I thought about Galileo and his work with acceleration of bodies and like your teacher i Agree that he too was a good hint.....From the E