A pendulum of mass m= 0.9 kg and length l=1 m is hanging from the ceiling. The massless string of the pendulum is attached at point P. The bob of the pendulum is a uniform shell (very thin hollow sphere) of radius r=0.4 m, and the length l of the pendulum is measured from the center of the bob. A spring with spring constant k= 12 N/m is attached to the bob (center). The spring is relaxed when the bob is at its lowest point (θ=0). In this problem, we can use the small-angle approximation sinθ≃θ and cosθ≃1. Note that the direction of the spring force on the pendulum is horizontal to a very good approximation for small angles θ. (See figure)
Take g= 10 m/s2
(a) Calculate the magnitude of the net torque on the pendulum with respect to the point P when θ=5∘. (magnitude; in Nm)
|τP|=
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(b) What is the magnitude of the angular acceleration α=θ¨ of the pendulum when θ=5∘? (magnitude; in radians/s2)
|α|=
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(c) What is the period of oscillation T of the pendulum? (in seconds)
T=
To calculate the magnitude of the net torque on the pendulum with respect to point P, we need to consider the torques due to gravity and the spring.
(a) Torque due to gravity:
The torque due to gravity will act in the clockwise direction. The formula for torque is τ = rF sinθ, where r is the distance from the pivot point to the point where the force is applied, F is the force, and θ is the angle between the force and the lever arm.
In this case, the lever arm is the length of the pendulum, which is l = 1 m, the force is the weight of the pendulum, which is mg, and the angle θ is 5 degrees. Therefore, the torque due to gravity is τg = l(mg)sinθ.
Substituting the given values: τg = (1 m)(0.9 kg)(10 m/s^2)sin(5 degrees).
(b) Torque due to the spring:
The torque due to the spring will act in the counterclockwise direction. The formula for torque is τ = rF sinθ, where r is again the distance from the pivot point to the point where the force is applied, F is the force, and θ is the angle between the force and the lever arm.
In this case, the lever arm is again the length of the pendulum, which is l = 1 m, the force is the force exerted by the spring, which is kx (where x is the displacement from the equilibrium position), and the angle θ is 5 degrees. Therefore, the torque due to the spring is τs = l(kx)sinθ.
Substituting the given values: τs = (1 m)(12 N/m)(x)sin(5 degrees).
The net torque on the pendulum with respect to point P is the sum of the torque due to gravity and the torque due to the spring:
|τP| = |τg| - |τs|.
To calculate the magnitude of the angular acceleration α=θ¨ of the pendulum when θ=5∘, we use the equation τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
In this case, the moment of inertia of the pendulum can be approximated as the moment of inertia of a uniform thin hollow sphere rotating about its diameter, which is (2/3)m(r^2 + l^2), where m is the mass, r is the radius of the bob, and l is the length of the pendulum. Therefore, I = (2/3)(0.9 kg)((0.4 m)^2 + (1 m)^2).
Given the torque |τP| and the moment of inertia I, we can use the equation to find the angular acceleration α.
To calculate the period of oscillation T of the pendulum, we can use the equation T = 2π√(l/g), where l is the length of the pendulum and g is the acceleration due to gravity.
Substituting the given values: T = 2π√(1 m/10 m/s^2).
To calculate the magnitude of the net torque on the pendulum (τP) when θ=5∘, we can use the formula:
τP = I α
where τP is the net torque, I is the moment of inertia of the pendulum, and α is the angular acceleration.
The moment of inertia for a thin hollow sphere of radius r rotating about an axis passing through its center is given by the formula:
I = (2/3) m r^2
where m is the mass of the pendulum and r is the radius of the hollow sphere.
(a) Calculate the magnitude of the net torque on the pendulum (|τP|):
|τP| = (2/3) m r^2 α
Given:
m = 0.9 kg
r = 0.4 m
α = θ¨ (angular acceleration)
We need to convert θ from degrees to radians before calculating the angular acceleration.
θ = 5∘ * (π/180) ≈ 0.0873 radians
Now, we can calculate the magnitude of the net torque by substituting the given values:
|τP| = (2/3) * (0.9 kg) * (0.4 m)^2 * α
Now that we have the equation, we can solve it for |τP|.