How to determine maximum amplitude of oscillation of a system, using the question below?
A large block P executes horizontal simple harmonic motion as it slides across a frictionless surface with a frequency f=1.50Hz. Block B rest on it and the coefficient of the static friction between the two is Us=0.600. what maximum amplitude of oscillation can the system have if block B is not to slip?
Solve this
To determine the maximum amplitude of oscillation of the system such that block B does not slip, you need to analyze the forces acting on the system.
The equation for the maximum amplitude of oscillation (Amax) can be derived using the condition when block B is on the verge of slipping. At this point, the static friction force between block P and block B reaches its maximum value and is equal to the force exerted on block B due to the horizontal acceleration of the system.
Step 1: Calculate the acceleration of the system.
Since block P is executing simple harmonic motion, its acceleration (a) can be given by the equation a = 4π²f²A. Remember that f represents the frequency and A represents the amplitude of the oscillation.
In this case, the frequency (f) is given as 1.50 Hz. Substitute this value into the equation to find the acceleration:
a = 4π²(1.50 Hz)²A
Step 2: Calculate the maximum static friction force.
The maximum static friction force (Ffriction) can be calculated by multiplying the coefficient of static friction (μs) by the normal force (N) between the two blocks. The normal force can be determined by considering the weight of both blocks. Since the blocks are on a frictionless surface, the normal force is equal to the weight of block B.
N = mB * g
Step 3: Equate the maximum static friction force to the force exerted on block B.
At the maximum amplitude of oscillation, the static friction force should just balance the force exerted on block B by the acceleration of the system.
Ffriction = mB * a
Step 4: Substitute the equation for the maximum static friction force with the equation for the acceleration and solve for the maximum amplitude (Amax):
μs * mB * g = 4π²(1.50 Hz)²Amax
Amax = (μs * mB * g) / (4π²(1.50 Hz)²)
Plug in the known values of the coefficients and masses to calculate the maximum amplitude of oscillation (Amax).