Defining Simple Harmonic Motion (SHM) AP Physics C Mechanics FRQ – Exam Style Questions etc.
Defining Simple Harmonic Motion (SHM) AP Physics C Mechanics FRQ
Unit 7: Oscillations
Weightage : 10-15%
Question
Note: the diagram shows a total length of 2L and an angle of θ from the vertical
A stick of length 2L and negligible mass has a point mass m affixed to each end. The stick is arranged so that it pivots in a horizontal plane about a frictionless vertical axis through its center. A spring of force constant k is connected to one of the masses as shown above. The system is in equilibrium when the spring and stick are perpendicular. The stick is displaced through a small angle θo as shown and then released from rest at t = 0 .
a. Determine the restoring torque when the stick is displaced from equilibrium through the small angle θo.
b. Determine the magnitude of the angular acceleration of the stick just after it has been released.
c. Write the differential equation whose solution gives the behavior of the system after it has been released.
d. Write the expression for the angular displacement θ of the stick as a function of time t after it has been released from rest.
Answer/Explanation
Ans:
a. F = –kx where for small angles x = Lθ (x = Lsinθ is also acceptable)
F = –kLθ
τ = FL cos θ ≈ FL = –kL2θ
b. τ = Iα where I = 2mL2
α = τ/I = – kθ/2m
c. α = d2θ/dt2
d2θ/dt2 = – kθ/2m
d. The equation above is the form of simple harmonic motion with an angular frequency ω = (k/2m)1/2
θ = θ0 cos (k/2m)1/2t