How would you find the displacement of a pendulum half way through its motion?

[canadianofpeace]

I understand that pendulum velocity is maximum at the equilibrium point ("bottom" of path) and that potential energy is greatest at the top. If we set 1/2mv^2=1/2kx^2, and solve for x and then divide by two, would that be the solution? According to TBR that is wrong but I can't wrap my head around why.

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[DrknoSDN]

Pendulum potential energy is equal to:
1/2 mgL(theta^2)

Similar to how length determines pendulum period the length determines the rate of conversion of maximum kinetic energy to a potential energy achieved at angle theta.

Conceptually. A 1 kg mass with Vmax of 5 m/s would reach a greater theta and "x" if the length of the pendulum is relatively short. If length is sufficiently long then x won't change significantly.

So if you convert half of your KE to PE it won't have a height of (0.5h). The KE = PE = mgh is true for complete conversions of energy.

 

[Czarcasm]

I understand that pendulum velocity is maximum at the equilibrium point ("bottom" of path) and that potential energy is greatest at the top. If we set 1/2mv^2=1/2kx^2, and solve for x and then divide by two, would that be the solution? According to TBR that is wrong but I can't wrap my head around why.

I may be mistaken here, but I'd calculate max velocity (where KE = max) and then find the height difference between initial and final (which is equal to the radius of motion), divide velocity by radius to get angular velocity and multiply it by total time it took to get to the middle to get angular displacement. But I'm not entirely sure this is even right since the velocity isn't constant. Actually, I know I'm wrong. There's probably some other equation you need to use, I think the one DrknoSDN gave for this specific situation (solve for theta). I need to brush up on Physics, sigh.

 

[NextStepTutor_1]

Also, note that while the equation you used for kinetic energy was correct, the equation for potential energy is the elastic potential energy. It only applies to springs and other materials that have elastic properties. The restoring force for pendulums is the force of gravity. Thus, in this case, you would use the gravitational potential energy equation and set it equal to kinetic energy so that:

mgh = 1/2 m*v^2

v = sqrt(2*g*h)

 

[DrknoSDN]

I understand that pendulum velocity is maximum at the equilibrium point ("bottom" of path) and that potential energy is greatest at the top. If we set 1/2mv^2=1/2kx^2, and solve for x and then divide by two, would that be the solution? According to TBR that is wrong but I can't wrap my head around why.

What study question were you working on?
Were you asking for the vertical displacement like "h" of mgh? Or the actual vector displacement? (a straight line from rest position to position at half arc)

Only reason I ask is because I tried to derive an equation directly relating velocity to a displacement vector between the rest position and the position of a pendulum with length L, when it has traveled half it's arc length.
That proved significantly more challenging than I initially thought. =/

If anyone want's to show off, throw down an equation for true displacement of a pendulum with velocity v, and length L. (mass m but I don't think it's required).

 

[Czarcasm]

What study question were you working on?
Were you asking for the vertical displacement like "h" of mgh? Or the actual vector displacement? (a straight line from rest position to position at half arc)

Only reason I ask is because I tried to derive an equation directly relating velocity to a displacement vector between the rest position and the position of a pendulum with length L, when it has traveled half it's arc length.
That proved significantly more challenging than I initially thought. =/

If anyone want's to show off, throw down an equation for true displacement of a pendulum with velocity v, and length L. (mass m but I don't think it's required).

I'm not 100% sure, but I believe you need to find angular displacement like you mentioned above (since it remains constant), and multiple it times the 'radius' or change in displacement to find the distance traveled. I just wish someone with more experience would confirm for us.