Pendulum's maximum momentum?

[Shindo]

Homework Statement

A pendulum hung by a string that is 90m (simplicity's sake) can come a maximum of 72m leftward or right ward. Where does it feel it's maximum momentum?

m=meters M=mass

Homework Equations

PE=mgh
KE=.5(M)(v)^2
Mv= momentum

The Attempt at a Solution

My question is: will a pendulum always have its maximum momentum right underneath its point of attachment (0 restoring force)? This is when Potential Energy is zero, so it should have it's maximum kinetic energy, meaning it's maximum speed and thus it's maximum momentum. Right?

So in finding the ball's maximum height, 90^2-72^2=54^2

Meaning it's 36 (90-54) meters above it's lowest point.

Mgh=M(9.8)(36)=353M

if we take the mass out of all equations (and still call it joules) then:

PE=353J

353J=.5(v)^2

v=26.6 m/s

Is this how you would find this kind of answer?

 

[Simon Bridge]

Reasoning from conservation of energy is the usual approach, yeah.
You cannot take the mass out of the equations and still call it joules though - it has to be called "square speed" or "meters per second all squared".

$$mgh = \frac{p^2}{2m} \implies p=m\sqrt{2gh}$$ ... so you cannot do it without knowing the mass.

 

[Shindo]

so you cannot do it without knowing the mass.

Right, but in this case since the mass doesn't change it doesn't change where the ball feels its maximum momentum.

 

[Simon Bridge]

Right, but in this case since the mass doesn't change it doesn't change where the ball feels its maximum momentum.

That's fine: the actual calculation you demonstrated from "attempt at a solution"? was finding the max speed of the pendulum. To find the max momentum you need the mass; in that calculation you wrote PE=353J ... which is false, actually PE=353(J/kg)M(kg); it is best practice to do the algebra before you put the numbers in.

The argument that max KE corresponds to max momentum is usually sufficient ... to prove it, mathematically, is different.
For that you want to find a relation between momentum and displacement, then find the maxima.

 

[HalfLostAndSearching]

Yes, your approach is correct. The maximum momentum of a pendulum will occur when it reaches its maximum speed at the bottom of its swing, right underneath its point of attachment. This is because at this point, the pendulum has the maximum kinetic energy, and momentum is directly proportional to kinetic energy. So, your calculation is correct and the maximum momentum of the pendulum in this scenario would be 353J.