Let's consider the solid disk to be of mass m and at the end of a rod.
The restoring torque on the rod of a torsion pendulum twisted at a small angle θ is expressed as:
, where k is the torsion constant.
This equation only applies to small θ.
The negative sign in the equation is because of the nature of restoring torque, which tends to oppose the twisting of the rod.
Now, the torque due to applied fore is expressed as:
, where r is the perpendicular distance between the center of rotation and the line of action of force.
The mass moment of inertia of a solid disc about the center of mass is expressed as:
, where m is the mass of the disc and R is the radius of the disc.
Natural frequency is expressed as:
The relationship between the pendulum's oscillation period (T) and the angular frequency (ω) is:
A torsion pendulum is made from a disk of mass m = 6.2 kg and radius R = 0.69 m. A force of F = 49.7 N exerted on the edge of the disk rotates the disk 1/4 of a revolution from equilibrium.
1. What is the torsion constant of this pendulum? answer N-m / rad
2. What is the minimum torque needed to rotate the pendulum a full revolution from equilibrium? answer N-m
3. What is the angular frequency of oscillation of this torsion pendulum? rad/s
4. Which of the following would change the period of oscillation of this torsion pendulum?
a. increasing the mass
b. decreasing the initial angular displacement
c. replacing the disk with a sphere of equal mass and radius
d. hanging the pendulum in an elevator accelerating downward
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