Classical Mechanics - Pendulum

Two pendulums are attached to a massless spring, as shown above. The arms of the pendulums are of identical lengths l, but the pendulum balls have unequal masses m₁ and m₂. The initial distance between the masses is the equilibrium length of the spring, which has spring contant K. What is the highest normal mode frequency of this system?

A.

B.

C.

D.

E.

(GR9677 #84)

Solution:

A. FALSE
ω = √(g/l) is angular frequency for single pendulum

B. and C are FALSE
Both answers do not depend on g/l

D. TRUE
If there is no dependence on K → ω = √(g/l)
And if m₂ → ∞,  m₁ will still oscillate with spring constant K but has no dependence on m_2, as if it were connected to a stationary object.

E. FALSE
If there is no dependence on K → ω = √(2g/l), not angular frequency for single pendulum
And if m₂ → ∞,  ω = √(2g/l) has no dependence on K and m₁

Answer: D 

Classical Mechanics - Pendulum

A cylindrical tube of mass M can slide on a horizontal wire. Two identical pendulum, each of mass m and length l, hang from the ends of the tube, as shown above. For small oscillations of the pendulums in the plane of the paper, the eigenfrequencies of the normal modes of oscillation of the system are 0,,   and …

A.

B.

C.

D.

E.

(GR9277 #07)

Solution:

2 modes in which the system can oscillate:

1. The 2 pendulum oscillate out of phase so there is torsional effect on the tube M → there is M in the equation.

2. In phase → no torsional effect on M → No M in the equation.

Answer: A

Classical Mechanics - Simple Harmonic Oscillation

Three masses are connected by two springs as above. A longitudinal normal mode with frequency ¹/_2π√^k/_m is exhibited by

1. A, B, C all moving in the same direction with equal amplitude.
2. A and C moving in opposite, and B at rest.
3. A and C moving in the same direction with equal amplitude, and B moving in the opposite direction with the same amplitude.
4. A and C moving in the same direction with equal amplitude, and B moving in the opposite direction with twice the amplitude.
5. None of the above.

(GR8677 #43)

Solution:

Angular frequency, ω = 2πf  → f = ^ω/_2π
For Simple Harmonic Oscillation (SHO), ω = √^k/_m
Thus, f = ¹/_2π√^k/_m is frequency of SHO.

SHO is a periodic motion of a particle about a fixed point (the centre of motion).

Only choice (B) gives a fixed point (mass B at rest).

Answer: B