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1. A block of mass m is at rest on top a platform and attached to a wall mounted (fixed) on the platform by means of a linear spring whose force constant is k. The platform is originally at rest and then set into harmonic oscillation
by a machine. and

are constants that are controlled by the machines settings. In addition, the block experiences a resistive force,
- bv, where b is a constant and v is the velocity of the block measured with respect to the platform.

Write down the differential equation that governs the motion of the particle in terms of x, the distance of the block measured from the wall. Indicate clearly whether you use an inertial or non-inertial frame in obtaining your answer.

Determine x(t) after the platform has oscillated for a very long time.

Determine for what setting of is the amplitude of x(t) a maximum for givenA, k, m, and b.





2. A particle is placed on top of a smooth (frictionless) hemispherical dome of radius R.

Identify the forces that are acting on the particle.

If the particle is slightly disturbed (i.e. given a negligibly small tangential velocity), at what point (as measured by in the figure) will the particle leave the sphere?

Derive the differential equation that governs)(twhile the particle is in contact with the dome.

If the particle is given a finite tangential velocity while it is at the top, determine the minimum velocity v0that the particle needs to have if it is to leave the sphere immediately (i.e. right from the top)?

With this initial velocity will the particle arrive at y = 0 without hitting the dome?




3. Two spaceships, S1 and S of mass m and m respectively, are moving in circular orbits in the same plane, of radius and with

and = , around the earth.

(a) Determine the periods of the two circular orbits.

Now the control center would like to send spaceship S1 to the orbit of Sto meet up with S. To accomplish this, the control center fires the rocket motor on S1 for a very short time to give S1 a larger tangential velocity so that it will go into an elliptic orbit arriving at orbit of Sat B. When S1 reaches the point B, the motor is fired again to bring the velocity to the necessary value for circular motion with radius .

(b) Determine the velocity of S1at A after the motor is fired, and the velocity at B before the motor is fired.

To meet up with spaceship S, both spaceships have to arrive at B at the same time.

(c) If at t = 0, both spaceships are observed to be along the same radial direction, how long should the control center wait before firing the motor of spaceship S1 to send it to the