# Physics 218: FINAL EXAM April 29 th, 2016

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2 Short Problems (40) Problem 2 (20) Problem 3 (20) Problem 4 (20) Problem 5 (20) Problem 6 (20) Problem 7 (20) Problem 8 (20) Problem 9 (20) Total Score (200) Short Problems (Circle the correct option) [NO Partial Credit] [40 Points] A) [8 points] A point-like particle is moving in one dimension along the x-axis. Its position is shown in the figure below as a function of time. What is the correct sign for the acceleration and velocity of the point particle at time t=to? (the arrows on the axes indicate the positive direction) i) a>0, v<0 ii) a<0, v>0 iii) a>0, v>0 iv) a<0, v<0 v) a=0,v>0 vi) a=0,v<0 vii) a>0, v=0 viii) a<0, v=0 2 of 12

3 B) [8 points] A simple pendulum consists of a point mass suspended by a massless, unstretchable string. If the mass is doubled while the length of the string remains the same, the period of the pendulum i) becomes 4 times greater ii) becomes twice as great iii) remains unchanged iv) becomes greater by a factor of 2 v) becomes 4 times smaller vi) becomes twice smaller vii) becomes smaller by a factor of 2 C) [8 points] A planet is orbiting a star. The planet s orbit is elliptical, such that the aphelion is twice as far away from the center of the star than the perihelion. The planet s speed is 40,000 m/s at the perihelion. What is its speed at the aphelion? i) 40,000 m/s ii) 20,000 m/s iii) 80,000 m/s iv) the answer depends on the planet s mass v) the answer depends on the planet-star distance at perihelion D) [8 points] An elevator is being lowered at a constant speed by a steel cable attached to an electric motor. Which statement is correct? i) The cable does positive work on the elevator and the force of gravity does positive work on the elevator. ii) The cable does negative work on the elevator and the force of gravity does negative work on the elevator. iii) The cable does negative work on the elevator and the force of gravity does positive work on the elevator. iv) The cable does positive work on the elevator and the gravity does negative work on the elevator. E) [8 points] A block of mass m slides to the right on a frictionless horizontal floor with speed V and collides inelastically head-on with a block of mass M, which is initially at rest. After the collision the two blocks move together with speed V/4. How does M compare to m? i) m = 2M ii) 2m = M iii) m =M iv) m=4m v) 4m = M vi) 3m = M vii) m = 3M 3 of 12

4 Problem 2 (20 points) An archer successfully hits a small target 50.0 m away (horizontal distance) at a height of 3.0 m above the ground. The arrow leaves the bow at a height of 1.7 m above the ground. The arrow s initial velocity makes 15 o angle with the ground. There is no wind and air resistance is negligible. a) What is the initial speed of the arrow? b) Now assume that the arrow is released from the ground (zero height). What is the maximum distance this arrow can cover in the horizontal direction before it hits the ground if it is released from the bow with the same initial speed as in part (a) and at an angle that provides for the longest (in horizontal direction) shot? 4 of 12

5 Problem 3 (20 points) A wrecking ball of mass M=500.0 kg is hung from a crane by a massless, unstretchable cord of length L=4.50 m. The ball is initially brought back such that the cord makes an angle θ=20 o with respect to the vertical and released from rest (see the figure below). a) What is the speed of the wrecking ball at the lowest point of its motion? b) What is the tension in the cord at this point? 5 of 12

6 Problem 4 (20 points) Two blocks on a frictionless horizontal surface with masses m1 and m2 are connected by a massless unstretchable string. The string can withstand a maximum tension force of Tmax. a) What is the maximum force F that can be exerted at an angle θ with respect to the horizontal on the block m1 without breaking the string? (Express your answer in terms of the known values θ, m1, m2, g, Tmax. Not all may be necessary.) b) Assume now that the force F is variable and is given as a function of time by the equation F(t) = A t, where A is a known constant. At t=0 s the blocks are at rest. What is the displacement of the block m2 at the time τ? (Express your answer in terms of the known values θ, m1, m2, g, Tmax, A, τ. Not all may be necessary. F(τ) < Tmax) 6 of 12

7 Problem 5 (20 points) A body of mass m slides at constant speed down an inclined plane that makes an angle θ with the horizontal. a) What is the coefficient of kinetic friction between the surface of the inclined plane and the body? (Express your answer in terms of the known values m, θ, g. Not all may be necessary.) b) The angle between the inclined plane and the horizontal is now increased to β (β>θ). What is the speed of the mass m at the bottom of the inclined plane after it slides down from height h starting from rest? The coefficient of kinetic friction between the surface of the inclined plane and the body is still the same as in part (a). (Express your answer in terms of the known values m, θ, β, h, g. Not all may be necessary.) 7 of 12

8 Problem 6 (20 points) The nucleus of an oxygen atom spontaneously decays from rest into two particles: an alpha particle (Mα = 6.64x10-27 kg) and the nucleus of a carbon atom (MC = 1.99x10-26 kg). The total kinetic energy of the system after the decay is 6.40x10-14 J. a) What is the speed of the center of mass of the system after the decay? b) What is the speed of the alpha particle after the decay? 8 of 12

9 Problem 7 (20 points) A massless, perfectly elastic, horizontal spring with a spring constant of 14,000 N/m is attached to the wall as shown. The other end of the spring is attached to a glider of mass 3.00 kg that is placed on top of a frictionless and horizontal air track. The system is initially at rest. At t=0s the glider is kicked so that it gains an instantaneous velocity Vo = 5.00 m/s in the -x direction. a) What is the maximum displacement of the glider from the equilibrium point? b) What is the displacement of the glider with respect to the equilibrium as a function of time? (Make sure to explicitly calculate amplitude, angular frequency and the phase angle.) 9 of 12

10 Problem 8 (20 points) A planet with radius 12,000 km makes one revolution on its axis in 20 hours. Objects located at the equator on the surface of this planet are in a state of apparent weightlessness (this means that they are orbiting the planet at 0 height without experiencing any normal force from the surface and remain motionless with respect to the planet s surface). a) What is the speed of these objects? b) What is the mass of this planet? 10 of 12

11 Problem 9 (20 points) A thin-walled hollow cylinder of radius R1 = 0.15 m and mass m1 = 0.35 kg is rotating with an angular velocity ω=2.00 rad/s. It comes into a contact with a solid cylinder of radius R2=0.25 m and mass of m2 = 0.45 kg that is resting on a frictionless surface. The first cylinder is dropped onto the second with their symmetry axes aligned (see the figure). Due to the force of friction between the cylinders they eventually rotate together without slipping. a) What is the final angular velocity of the two-cylinder system? b) What energy was released as heat? 11 of 12

12 Extra Space make sure you indicate on the main page of the problem that you are continuing here 12 of 12

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