1. Why is the velocity of the mass of the pendulum equal to to sqrt (rg tan thet
ID: 2143058 • Letter: 1
Question
1. Why is the velocity of the mass of the pendulum equal to to sqrt (rg tan theta) ? Derive this relationship using materials about conical/circular pendulums.
2. What happens to the period of the pendulum as the radius increases? What happens as the length increases? (Check how the angle varies, and then see how the period varies.) Is the pendulum DIRECTLY proportional or INDIRECTLY proportional to r? To L? to Theta?
3. Take the partial derivative of the period of the pendulum with respect to theta and with respect to r.
4. Is the pendulum dependent on L? There are a few different ways of looking at this.
Explanation / Answer
A change of mass would not effect a pendulum. The easiest way to explain is by the kinetic and potential energy. Potential energy is mass * height and kinetic energy is 1/2*mass*velocity squared. As the amount of energy is constant in the system the potential energy would equal kinetic energy (mh=.5mv^2). Notice that there is the variable m on both sides--due to algebraic laws the ms on both sides cancel. Therefore h=.5v^2 so velocity is dependent only on the original starting height.
The term period refers to the length of time to complete a circuit so the faster the pendulum is going the smaller the period. As velocity is dependent only on the starting height; mass has no effect on a pendulum.
You can model a pendulum as a mass m on the end of a weightless string length l. If the angle is A from the vertical, resolving forces radially:
-m x l x d^2A/dt^2 = m x g x sinA
d^2A/dt^2 = -(g/l) sinA
Provided the angles are small, sinA approximately equals A. Therefore
d^2A/dt^2 = -(g/l)A
This is simple harmonic motion (acceleration directly proportional to displacement).
angular velocity = w^2 = (g/l)
So time period = T=2pi/w = 2pi x root(l/g)
[T equals the time for the pendulum to swing back and forth]
So at small angles the mass has no affect on the time period, however, the length of the pendulum does!
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