1). A mass on a spring oscillates with a period of 2s. It's maximum displacement is 0.25m. The mass is at x=0 at t=0. (Assume there is no damping.).
a). Find the equation of motion.
b). If the total energy is 0.1J, what is the mass and what is the spring constant?
2). A simple harmonic oscillator has a mass M0 on a spring of force constant k. The system is damped. It takes 10 cycles for the amplitude of the oscillations to decay to 37% of the original value (i.e. 1/e).
a) If I double the mass to 2M0 how many cycles will it take to decay to 1/e? (assume b in F(damping)=(-b*v), stays the same.) (Hint: Look at what happens to 'beta' and omega when you change the mass.)
b). I change the mass, but leave the damping force, (-b*v), the same, i.e. b is the same for both masses. If it is now critically damped, what is the new mass M1 in terms of M0?
3). A particle of mass M initially is at rest on a smooth (frictionless) plance a distance B from the origin. The plane initially makes an angle theta=0 with the horizontal, i.e. it is flat, but theta increases at a constant rate, omega0 until it reaches theta<90 degrees.
a). Find the equation of motion for the particle while theta is increasing.
b). Solve for position as a function of time.