Moments & Pendulum: Different Lengths, Same Oscillation?

[sgstudent]

if i have two forces acting at two different angles at a same spot will the addition of the moments of the two forces be the same as the moment whereby i take the net force of the two forces and get the moment of the net force of the same two forces?

is there a way to explain why for the same mass of the bob, two different lengths of the pendulum will have different times to have a complete oscillation using moments to explain it. Thanks for the help!

 

[tiny-tim]

hi sgstudent!

if i have two forces acting at two different angles at a same spot will the addition of the moments of the two forces be the same as the moment whereby i take the net force of the two forces and get the moment of the net force of the same two forces?

yes … r x a + r x b = r x (a + b)

is there a way to explain why for the same mass of the bob, two different lengths of the pendulum will have different times to have a complete oscillation using moments to explain it.

yes, the moment tells you the angular acceleration

 

[sgstudent]

hi sgstudent!


yes … r x a + r x b = r x (a + b)


yes, the moment tells you the angular acceleration

Thanks tiny Tim. However, I don't really know the explanation using moments on why a longer pendulum will have longer periods than shorter ones. Since they are longer so won't their moment be greater? Thanks!

 

[HallsofIvy]

The longer the pendulum, the longer the distance it has to cover to complete one period. Why would you think the time to cover this longer distance would be shorter?

If you want a detailed answer:
Suppose the pedulum has mass m centered at distance L from the pivot. There is a downward force of strength -mg. But the pendulum mass can only move around the circumference of the circle or radius L and the component of force parallel to the circumference is $-mg sin(\theta)$ where $\theta$ is the angle the pendulum makes with the vertical.

If we measure $\theta$ in radians, the angle $\theta$ corresponds to a distance around the arc of $s= L\theta$ and so the linear velocity is $v= ds/dt= L d\theta/dt$ and the acceleration is $a= dv/dt= L d^2/theta/dt^2$. Since "mass times acceleration= force", the motion is given by
$$ma= mL d^\theta/dt^2= -mg cos(\theta)$$

That is a badly "non-linear" equation so there is no simple exact solution but there are a number of ways to approximate it. One is to note that for small angles, $cos(\theta)$ is approximately $\theta$ itself so we can approximate the equation by
$$mLd^2\theta/dt^2= -mg\theta$$

That is a second order linear equation with constant coefficients. It has "characteristic equation" $Lr^2=-g$ which has "characteristic roots" $\pm\sqrt{g/L}$. That, turn, tells us that two independent solutions are $sin(\sqrt{gt/L}$ and $cos(\sqrt{gt/L}$. That will have period given by $gt/L= 2\pi$ so that $t= 2L/g$ which is directly proportional to L- the larger L, the longer the period.

 

[PJC01]

I can confirm that the addition of the moments of two forces acting at different angles at the same spot will not be the same as the moment of the net force of the same two forces. This is because moments are calculated by multiplying the force by the distance from the pivot point, and the distance will be different for each of the two forces.

Regarding the question about the different lengths of a pendulum having different times for a complete oscillation, this can be explained using moments. The time for a complete oscillation of a pendulum is dependent on the length of the pendulum and the acceleration due to gravity. The longer the pendulum, the larger the moment of inertia, which means it takes longer for the pendulum to complete one oscillation. This is because the longer pendulum has a larger distance from the pivot point, resulting in a larger moment of inertia.

Additionally, the moment of inertia also affects the period of oscillation of the pendulum. A longer pendulum will have a longer period of oscillation compared to a shorter pendulum with the same mass. This can be explained by the equation T=2π√(I/mgL), where T is the period, I is the moment of inertia, m is the mass, g is the acceleration due to gravity, and L is the length of the pendulum.

In summary, the different lengths of a pendulum will have different times for a complete oscillation due to the differences in their moments of inertia, which is dependent on the length of the pendulum. I hope this explanation helps.