How Much Force and Speed Does a Motor Need to Maintain Pendulum Motion?

[starcrossed]

I had earlier posted a question in this section, but it got moved to homework section. so i did some homework and am posting the question for some more help.

I am interested in determining the force required to be supplied to a pendulum to overcome its damping force, so that the pendulum can be kept moving.

These are the steps i followed.

1. Determing the potential energy of the pendulum.

The mass of pendulum is m=10 kg. the length of the pendulum is L=0.30 m. i am raising the pendulum by angle A=20 degrees and releasing it. The potential energy of the pendulum at 20 degrees angle of swing is

E=mgh= mgl(1-cosA)=10*9.81*0.30(1-cos(20))= 1.7733 joules.

2. i release this pendulum and it comes to rest in 50 oscillations. this total energy loss in 50 oscillations is energy loss due to friction and air resistance forces combined to gether. i will call it damping energy loss.

3. if i assume linear loss of energy the damping loss of energy per oscillation is 1.7733/50 = 0.036 joules.

4. however this is wrong assumption as the energy loss is exponential with highest energy loss in the 1st oscillation and the least energy loss in the last oscillation.

5. To calculate the max energy loss due to damping: I start the oscillation at 20 degrees angle of swing. after the first oscillation the angle of swing decrease to say 19 degrees. the potential energy of the pendulm at beginning with 20 degrees angle of swing is 1.7733 joules.
E=mgh= mgl(1-cosA)

After the 1st oscillation when the pendulum returns to the same position the angle is 19 degrees. the pe at 19 degrees is 1.60. E=mgh= mgl(1-cosA)

hence the loss energy due to damping is 0.17 joules.

so if i have to keep the pendulum in motion, i should supply 0.1 joules of energy to the system.

QUESTION:

now this energy is to be supplied to the pendulum by kicking or pulling the pendulum in its direction of motion. This is the max energy that has to be supplied to pendulum to keep it moving at the same amplitude and frequency with which it started the oscillation.

now for eg we pull this pendulum with a motor, where a small string is tied with the shaft of the motor and this motor pulls the pendulum from the bottom.

my question is that what should be the torque and speed of this motor? how do i determing what force i have to apply and at what speed so that the required energy of 0.1 joules is supplied to the system?

i mean if i am using a motor to pull this swing...at what speed this pendulum be pulled? should the motor be rotated in such a way that it pulls the pendulum at twice the speed of its oscillation or 1.2 times the speed of its oscillation..? how do i determing this ?

 

[gtoguy97]

i know the basics of torque and speed of motor but what exactly i have to do for this problem is not clear to me. please help.

 

[astrokreng]

I would like to commend you for your thorough approach in determining the energy requirements for keeping a pendulum in motion. Your understanding of the potential energy and damping forces involved is impressive.

To answer your question, the torque and speed of the motor needed to supply the required energy to the pendulum will depend on various factors such as the efficiency of the motor, the mass and length of the pendulum, and the amount of damping force present.

One way to approach this problem is to use the conservation of energy principle. The energy supplied by the motor should be equal to the energy lost due to damping. So, if the damping energy loss per oscillation is 0.036 joules, the motor should supply the same amount of energy per oscillation to keep the pendulum in motion. This means that the motor should rotate with enough torque and speed to supply 0.036 joules of energy in the time it takes for one oscillation.

To determine the required torque and speed, you can use the formula for rotational kinetic energy, which is 1/2 * I * ω^2, where I is the moment of inertia and ω is the angular velocity. The moment of inertia of a pendulum can be calculated using the formula I = ml^2, where m is the mass and l is the length of the pendulum.

Using this formula, you can calculate the required angular velocity of the motor to supply 0.036 joules of energy in one oscillation. The torque of the motor can then be calculated using the formula τ = I * α, where α is the angular acceleration. The motor should rotate with enough torque and speed to supply the required energy in one oscillation.

In terms of the motor speed, it would be best to match the speed of the pendulum's oscillation to minimize energy losses and maintain a constant amplitude and frequency. However, this may not always be feasible depending on the motor's capabilities. In general, the motor should rotate at a speed that is close to the pendulum's natural frequency, which can be calculated using the formula f = 1/2π * √(g/l).

I hope this helps you in determining the required torque and speed of the motor to keep the pendulum in motion. Keep up the good work!