Simple pendulum dynamics; equations of motion, work and energy

In summary, the conversation discusses a physics problem involving a pendulum and the use of Euler's laws of motion and work and energy equations to find the tension, angular acceleration, and angular velocity. The person is unsure if they made any mistakes in their calculations and is seeking help on how to plot the tension in relation to time. They are advised to consider the conservation of energy in their approach.
  • #1
xzibition8612
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Homework Statement



See attachment "problem"

Homework Equations


Euler's laws of motion (moment equations), work and energy equations


The Attempt at a Solution


See attachment "work"

I did the work for (1) and (2). I end up with two equations: the first is the tension T, the second is the angular acceleration. I'm not so sure if I made any mistakes in solving the equations of motion, but I'm not really comfortable with these two equations and feel like something went wrong. I just can't see it. Remember the pendulum is a point mass, hence for the point mass the moment of inertia is ml^2. Now for question (3), how do I find the angular velocity? None of my two equations contain this quantity, hence I feel like something went wrong. For (4), I do indeed have T= 9.83/sin(theta). Now how do I plot this in relation to time? Because from this equation I can only plot the tension T in relation to the change in angle, not time. I'm not going to worry about (5) right now, got to get (1)-(4) right first.

Thanks for the help.
 

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  • #2
Perhaps look at it from the point of view of energy eg KE at the bottom = PE at the top.
 

FAQ: Simple pendulum dynamics; equations of motion, work and energy

What is a simple pendulum?

A simple pendulum is a weight suspended from a fixed point by a string or rod. It is considered a simple harmonic oscillator, meaning its motion follows a predictable pattern.

What are the equations of motion for a simple pendulum?

The equations of motion for a simple pendulum are:
- The equation for displacement:
θ = θ0cos(√(g/L)t)
- The equation for velocity:
v = -θ0√(g/L)sin(√(g/L)t)
- The equation for acceleration:
a = -θ0(g/L)cos(√(g/L)t)
Where θ0 is the initial displacement, g is the acceleration due to gravity, and L is the length of the pendulum.

How is work related to a simple pendulum?

Work is the amount of energy transferred to or from an object by a force. In the case of a simple pendulum, work is done by the force of gravity as the pendulum swings back and forth. The work done by gravity is equal to the change in potential energy of the pendulum as it moves from one point to another.

What is the relationship between energy and a simple pendulum?

A simple pendulum has both kinetic energy (energy of motion) and potential energy (energy due to position). As the pendulum swings, its energy is constantly changing between kinetic and potential. At the highest point of the swing, all of the energy is in the form of potential energy, and at the lowest point, all of the energy is in the form of kinetic energy.

How does the length of a simple pendulum affect its dynamics?

The length of a simple pendulum affects the period (time for one complete swing) and the frequency (number of swings per unit time). A longer pendulum will have a longer period and a lower frequency, while a shorter pendulum will have a shorter period and a higher frequency. The length also affects the amplitude (maximum displacement) of the pendulum, with longer pendulums having larger amplitudes.

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