Physics 3210, Spring 2017 Exam #1 Name: Signature: UID: Please read the following before continuing: Show all work in answering the following questions. Partial credit may be given for problems involving calculations. Be sure that your final answer is clearly indicated, for example by drawing a box around it. Be sure that your cellphone is turned off. Your signature above indicates that you have neither given nor received unauthorized assistance on any part of this exam. Thanks, and good luck!
1. (8 pts) A locomotive engine is pulling three boxcars behind it, each having equal mass M. The cars are connected by cables numbered 1, 2 and 3. The train has acceleration a to the right. Determine the tension in each of the three cables.
2. (8 pts) A spider is standing on an ice cube, which is sliding in a circular bowl of radius R. At the bottom of the bowl, the normal force of the ice cube on the spider is equal to three times the spider s weight. What is the speed of the ice cube at the bottom of the bowl? 3. (8 pts) A cannon fires a projectile with initial speed v at an angle of θ with respect to horizontal. The projectile flies under the influence of uniform gravity, before returning to Earth. Assume air resistance is negligible. Indicate with a T or F whether the following statements are true or false: (a) (b) (c) (d) (e) (f) (g) (h) The acceleration of the projectile remains constant during its upward flight. The vertical (y) component of the projectile s velocity remains constant. The y component of the projectile s velocity is never zero. The horizontal (x) component of the projectile s velocity remains constant. The x component of the projectile s velocity is never zero. The minimum speed of the projectile during its flight is equal to vsinθ. The horizontal acceleration of the projectile is zero. The vertical acceleration of the projectile is zero.
4. (14 pts) A tetherball (see picture) with mass of 1.25 kg rotates in a perfect circle. A massless cord connecting the ball to the pole has length L, and the cord makes an angle θ = 35 with respect to the pole when the speed of the ball is 2.80 meters/second. (a) (8) What is the length L of the cord? (b) (6) Calculate the time for the tetherball to make one orbit, at this speed.
5. (12 pts) A torsion pendulum consists of a rod or wire hanging from the ceiling and a bob which rotates about the axis of the rod rather than swinging from side to side. In the figure below, the bob is a flat disk. As we ll derive later in the course, the rotation angle θ of the bob satisfies the differential equation: d 2 θ(t) dt 2 + 2k mr 2θ(t) = 0 where m is the mass of the bob, R is its radius, and k is the torsion constant which plays a role similar to the spring constant k for the linear harmonic oscillator. For the problems below, assume m = 0.010 kg and R = 0.05 meters. (a) (4) The period of oscillation of the disk is 0.70 seconds. What is the torsion constant?
(b) (8) If the disk is initially at θ = 0 and is given an initial angular speed of 1.0 radians/s, find the particular solution to the differential equation by evaluating all constants of integration.
6. (18 pts) A block of mass m = 0.75 kg rests on top of a block of mass M = 2.0 kg. A string attached to the block of mass M is pulled so that its tension is F T = 6.0 N at a 30 angle to the horizontal as shown. The blocks move together. The coefficient of static friction at the surface between the blocks is µ s = 0.25; there is no friction at the surface between block M and the floor. (a) (6) What is the acceleration of the two-block system? (b) (4) What is the direction of the frictional force being exerted on the top block m?
(c) (8) The tension F T is now increased. What is the maximum tension F Tmax with which the string can be pulled such that the blocks continue to move together (i.e. that the block of mass m does not start to slide on top of the block of mass M)?
7. (18 pts) An satellite orbits the Earth at a constant distance R o from the center of the Earth, at a constant speed v. Useful Constants: G = 6.673 10 11 m 3 /(kg s 2 ) M E = 5.972 10 24 kg R E = 6.371 10 6 m (a) (6) Write down the general expression for the velocity vector v and acceleration vector a in polar coordinates. (b) (6) Simplify the expressions for v and a, for the satellite s uniform circular motion.
(c) (6) Use the above to evaluate the velocity of the satellite as a function of G, M E, and R o. Derive a numerical answer (in kilometers/second) for the case R o = R E.
8. (14 pts) Two masses are connected as shown in the figure below. M 1 is initially at rest on a ramp with coefficient of static friction µ s. M 2 is connected to M 1 via a massless, frictionless pulley and a massless string. (a) (6) Draw the free body diagrams of masses M 1 and M 2. (Assume the friction force is acting so as to prevent M 1 from sliding up the ramp.)
(b) (8) In terms of M 2, θ, and µ s, find the minimum value of M 1 such that it will not slide up the ramp.