Physics --- Vibrations and Waves -- I really need help with these!!!?

Question

Please me:

1. Real springs have mass. How will the true period and frequency differ from those given by the equations for a mass oscillating on the end of an idealized massless spring?

2. How could you double the maximum speed of a SHO (Simple Harmonic Oscillator)?

For these problems, please show and explain your work so that I can understand this topic in Physics better! Some answers from the back of the book are listed but I can't seem to get them right.

1. When a 65kg. person climbs into a 1000kg car, the car's springs compress vertically by 1.9 cm. What will be the frequency of vibration when the car hits a bump? Ignore damping.
(0.89 Hz)

2. A piece of rubber is 50 cm long when a weight of 35 N hangs from it & is 85 cm long when a weight of 70 N hangs from it. What is the "spring" constant of this piece of rubber?

3. If a particle undergoes SHM w/ amplitude .15m, what's the total distance it travels in 1 period.

Answer 1

1. The added mass in a real spring will reduce the natural frequency (increase the period). Since it is a distributed mass, not a lumped mass, the quantitative effect of this added mass is not a simple expression.
2. To double the speed you can double the frequency while keeping the amplitude constant, or you could double the amplitude while keeping the frequency constant. How do you double the frequency? Quadruple the spring stiffness or cut the mass to 1/4. Remember, the radian frequency ^ 2 = K/M.
Problems:
1. The total spring constant of the car K = 65 kg * g / 1.9 cm, expressed in N/m. The vibration frequency in rad/s = sqrt(K/M). You have K, and the answer appears to be for the total (car + person) mass. Go.
2. Spring constant is force / deflection = 35/35 N/cm.
3. The amplitude is the distance from 0 to maximum deflection A. In one period the particle goes from 0 to A to 0 to -A to 0. That's a total distance of 4*A.