Pendulum mass and Newton's second law

[moomoocow]

hell
i have a question:
if i changed the mass of the ball on the pendulum, i assume that Changing the mass of the ball would not change the period of the pendulum because gravity pulls objects towards Earth at the same speed regardless of their mass.
however
Newton's second law says that the more mass, the slower the acceleration:
which means that changing the mass of the ball on the pendulum will change the period of the pendulum.

please tell me which explanation is correct:

 

[quasar987]

The first.

The period of oscillation of a frictionless pendulum for small oscillation about the equilibrium is

$$T=2\pi \sqrt{\frac{l}{g}}$$

where l is the length of the string and g the gravitationnal acceleration.In an analogy to the mass-spring system, g is the analogue of the spring constant k and l is the analogue of the mass m, because the period of oscilattion of the mass-spring system is

$$T=2\pi \sqrt{\frac{m}{k}}$$

So if you wanted to affect the period of oscillation of a pendulum in the same way as changing the mass of a mass-spring system affects the period of oscillation, you'd have to change the length of the string by the same amount as you change the mass.

 

[moomoocow]

thank you..
so to change the period of the pendulum, we would have to change the length of the string

 

[quasar987]

yes, or move to a different gravitational field

 

[HallsofIvy]

Newton's second law: F= ma, says that for a fixed force the greater the mass, the slower the acceleration.

However, Newton's law of gravity: $F= -\frac{GmM}{r^2}$ says that the greater the mass the greater the gravitational force.

Putting those together
[tex]-\frac{GmM}{r^2}= ma[/itex]<br /> the "m"s cancel so <br /> $$a= -\frac{GM}{r^2}$$<br /> independent of the mass of the object.[/tex]

 

[HallsofIvy]

yes, or move to a different gravitational field

Which can happen (slightly) if you move from sea level to a high mountain.

Most grandfather clocks (I have one) have a screw for moving the weight up or down the pendulum bar, thus controlling the speed of the clock.