Simple Pendulum and harmonic motion

[Husker70]

Homework Statement

A sim[le pendulum has a mass of .25kg and a length of 1.0m. It is displaced
through an angle of 15degrees and then released. What are (a) the maximum
speed, (b) the maximum angular acceleration, and (c) the maximum restoring
force? Solve the problem once by using the simple harmonic motion model for
the motion of the pendulum, and then solve the problem more precisely by
using more general principles.

Homework Equations

The Attempt at a Solution

The problem that I am having is getting started. I know this isn't
hard. I got the part (c) for the second on with the restoring force to be
mgsin theta = .634N and it is correct. I know the answer for the simple
models are close to the more precise ones but I can't find any way to
start.
Thanks for any help,
Kevin

 

[phyzmatix]

Think of where the bob will reach its maximum speed. You might also want to make use of the law of conservation of energy

 

[thomate1]

I would approach this problem by first reviewing the principles of simple harmonic motion and the equations that govern it. In the case of a simple pendulum, the restoring force is given by F = -mg sinθ, where m is the mass, g is the acceleration due to gravity, and θ is the angle of displacement from the equilibrium position.

Using this equation, we can determine the maximum restoring force by substituting the given values of mass (0.25 kg) and angle (15 degrees) into the equation: F = -(0.25 kg)(9.8 m/s^2) sin(15 degrees) = 0.634 N. This confirms the answer for part (c) obtained using the more general principles.

To find the maximum speed, we can use the equation v_max = Aω, where A is the amplitude (in this case, the maximum displacement from the equilibrium position) and ω is the angular frequency. Since the pendulum has a length of 1.0 m, the amplitude can be calculated by A = Lθ = (1.0 m)(sin 15 degrees) = 0.26 m. The angular frequency can be found using the equation ω = √(g/L) = √(9.8 m/s^2 / 1.0 m) = 3.13 rad/s. Thus, the maximum speed is v_max = (0.26 m)(3.13 rad/s) = 0.82 m/s.

Finally, the maximum angular acceleration can be determined using the equation α_max = -ω^2A = -(3.13 rad/s)^2(0.26 m) = -2.56 rad/s^2.

In summary, the maximum speed of the pendulum is 0.82 m/s, the maximum angular acceleration is -2.56 rad/s^2, and the maximum restoring force is 0.634 N.

I would also mention that while the simple harmonic motion model provides a good approximation for the motion of a pendulum, it does not take into account factors such as air resistance and the mass distribution of the pendulum. Therefore, for more precise calculations, it is important to use the more general principles of motion.