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The Simple Pendulum
10
Yay you made it!
Hopefully you've gained a better understanding of simple pendulums :)
Now in plug the values that we know:
= 0.400kg*9.8 m/s^2 *0.30m*(1-cos(4*2π/360))
= 2.86*10^-3 J
*Note we multiplied 4 degrees by 2π/360 to turn it from degrees to radians.
L-Lcosθ
Q2 What is the kinetic energy of the pendulum at that time?
Based on the law of conservation of energy
Ekmax = Epmax = mgh = mg(L-Lcosθ)
= mgL(1-cosθ)
L-Lcosθ
The Simple Pendulum
Relating this back to the mass spring systems and the equation for simple harmonic motion:
w = √(k/m) , where the spring constant represented the restoring force and the mass influenced oscillation time
In this case w = √(g/L) where gravity acts as the restoring force and the length influences the oscillation time
We know that the period is T = 2π/w = 2π√(L/g)
Since we know that the highest speed is reached after ¼ of the period:
t = 1/4T = 1/2 π√(0.30/9.8) = 0.27 s
Since the pendulum is at its highest speed when its angle is 0, the pendulum first reaches this speed at the first quarter of the period/ oscillation.
Let's start with Q1
How much time does it take the pendulum bob to reach its highest velocity?
Based on your own experience, if you have ever seen a pendulum, it seems to be moving fastest at the bottom position when the angle is 0.
This can be shown by the graph below, representing the pendulum movement in simple harmonic motion. One full oscillation/ period occurs when the pendulum bob returns to the spot where it started. The first point where the pendulum bob reaches its highest velocity is indicated by the green circles. Notice how the graph starts off by going at a negative velocity. This is because the pendulum starts at the right, so it undergoes negative velocity to the left.
*Note the graph is only a representation of the movement of the pendulum, and the values given are not accurate
*Taken from the PHYS 101 textbook (revised custom volume 1
and annotated
What information do we have:
200 meters ahead
Angle of pendulum: 4 degrees
Length of string: 0.30 m
Mass of pendulum: 400g
*Taken from www.math.duke.edu and annotated
A simple pendulum is pulled to the right at a 4 degree angle and released from rest. The length of the string is 0.30 m and the mass of the pendulum is 400g.
(1) How much time does it take the pendulum bob to reach its highest velocity?
(2) What is the kinetic energy of the pendulum at that time?