Q.1 Grading: (a) 10, (b) 4, (c) 6 points A uniform beam 9.00 m long, weighing 300 N, rests symmetrically on two supports 5 m apart. A boy weighing 600 N starts at point A and walks toward the right.

(a) Calculate the upward forces AF and BF exerted on the beam at points A and B, as

functions of the coordinate x of the boy. Draw the graphs of )(xFA and )(xFB on the same diagram below.

(b) How far beyond point B can the boy walk before the beam tips? (c) How far from the right end of the beam should support B be placed so that the boy can walk just to the end of the beam without causing it to tip?

(m) x

(N))( and xF(x)F BA

400

600

0 2 4 6 82−

800

200

Q.2 Grading: (a) 10 (b) 10 points A bullet with mass m and a velocity of v is shot perpendicularly to the edge of a rod of length L and mass mM 2= , which is free to rotate on a flat frictionless surface about a pivot at point P, as shown in the figure. The bullet is lodged into the rod and they move together after the collision. (a) Find the angular velocity ω of the rod + bullet after the collision. (For the rod 2

121

.. MLI mc = )

(b) How much kinetic energy is lost in this collision? Express your answer in terms of m and v.

P

Top view

3/L

v

m2

m

Q.3 Grading: (a) 12 (b) 8 points Two blocks, with identical masses m each, are connected by a string of negligible mass passing over a pulley of radius R and mass m as shown in the figure. (a) Find the acceleration a of the system (assuming the inclined plane is frictionless, and the string does not slip over the pulley). (b) Determine the tensions 1T and 2T in the two parts of the string.

(For the pulley 221

.. mRI mc = )

2T

30

m

mm

R1T

a

Q.4 Grading: each part is 5 points A body of mass m is moving along the x-axis. Its potential energy is 2222 )()( axbxU −= . (a) What is the force )(xF acting on the body? (b) Find the locations of all equilibrium points. (c) The body starts with zero speed at ax 4= . What is its speed at 0=x ? (d) Sketch the potential energy graph (U versus x) and determine which of the points found in part (b) correspond to a stable equilibrium.

Q.5 Grading: each part is 5 points A pendulum of length L and mass m has a spring of force constant k connected to it at a distance h below the point of suspension as shown in the figure. Assume that the vertical suspension rod is rigid, but ignore its mass. The system is displaced from its equilibrium position by an angle θ and released. (a) Draw the free body diagram to locate the forces acting on the pendulum during its motion. (b) Calculate the torque due to these forces about the point of suspension. (c) Show explicitly that the torque is proportional to θ (for small values of θ ). (d) Use the rotational analog of Newton's second law to calculate the frequency of oscillations.

m

L k

h