Simple Harmonic Motion (mostly)

1.

Consider a pair of coaxial springs. The spring constant of the outer (h₁ = 1.2 m high) spring is k₁ = 800 N/m and that of the inner (h₂ = 0.7 m high) spring is k₂ = 1200 N/m. A 90 kg mass is placed on top of this system and allowed to descend gradually to a state of static equilibrium.

(a)

Where is the static equilibrium position?

(b)

What is the period of a 1 cm oscillation about equilibrium?

2.

A frictionless roller coaster is built in a special shape such that if  x  is the horizontal distance from the centre (low point) of the parabola and  y  is the height above the low point,  x(t)  will exhibit ${\cal SHM}$. What is the shape  y(x)?

3.

A 1000 kg car sits on a frictionless level surface attached to a horizontal spring. What spring constant will cause it to oscillate with a period of 1 s?

4.

Two identical masses hang from a spring. Previous measurements have shown that the period of oscillation of this system is 1.0 s. By damping out all oscillations, we prepare the system in a state of equilibrium, for which the distance  x₀  from the ceiling to the equilibrium position of the end of the spring is 1.0 m. The string between the two masses is then cut.

(a)

What is the new period of oscillation?

(b)

What is the new equilibrium position?

(c)

What is the amplitude of the oscillation?

5.

Captain Picard places the starship Enterprise in a circular orbit about a 1-Solar-mass neutron star (radius 10 km) at a safe (??) distance of 6,000 km from its centre.

(a)

What is the centripetal acceleration of the centre of gravity of the Enterprise?

(b)

What is the period of its orbit?

(c)

If Picard's head is 1.6 m further from the star than his feet, what is the difference between the gravitational accelerations of his head and feet?

(d)

Comment on the probable effect of this difference.

6.

For a light mass m in a circular orbit about a much heavier mass M, prove that the total energy  E = K + U  is half the gravitational potential energy  U,  where we adopt the usual convention $U \to 0$ as $r \to \infty$. (K is the kinetic energy.)