Mechanics: Dynamics (Translational and rotational motion).

1. Two equal weights of mass M each are connected by a cord passing over a frictionless pulley of mass mp and radius r. An additional weight of mass m is placed on one of the weights of mass M. Derive the expression for: a) system acceleration a; b) cord tensions.

2. A helicopter rotor blade can be considered a long thin rod of mass m and length l. Derive the expression for the moment of inertia of the three rotor blades as well as for the torque the motor must apply to bring the blades up to speed of w [rev/s] in t seconds.

3. A ring is rolling on a horizontal surface at speed v0 when it reaches a theta- incline. How far along the incline will it go? How long will it be on the incline before arriving back at the bottom?

4. A uniform rod of mass m and length L can pivot freely (without friction) about a hinge attached to a wall. The rod is held horizontally and then released. Determine: a) the angular acceleration of the rod at the moment of release.

5. The radius of the roll of paper is r = 7.6cm and its moment of inertia is I = 2.9x10-3 kg*m2. A force F = 3.2 N is exerted on the end of the roll for t = 1.3 s, but the paper does not tear so it begins to unroll. A constant friction torque Mf = 0.11 m*N is exerted on the roll which gradually brings it to a stop. Assuming that the paper's thickness is negligible, calculate: a) the length of paper that unrolls during time that force is applied (t=1.3 s) and b) the length of paper that unrolls from time the force ends to the time when the roll stopped moving.

6. A wheel of mass m has radius r. It's standing vertically on the floor and a horizontal force F is exerted on the wheel's axle so that it will climb a step of height h (h<r) against which the wheel rests. What minimum force is needed?