MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.01 Physics Fall Term = # v x. t " =0. are the values at t = 0.

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.01 Physics Fall Term 2012 Exam 1: Practice Problems! d r!! d v! One-Dimensional Kinematics: v =, a = dt dt t " =t v x (t)! v x,0 = # a x ( t ") d t ", x(t)! x 0 = # v x ( t ") d t" t " =0 Initial conditions: x 0 and v x,0 are the values at t = 0. t " =t t " =0 Newton s Second Law: Force, Mass, Acceleration:!! Newton s Third Law: F1,2 =! F2,1! F = m! a Peter Dourmashkin 2012

2 Problem 1: Elevator A person of mass m p stands on a scale in an elevator of mass m p. The scale reads the magnitude of the force F exerted on it from above in a downward direction. Starting at rest at t = 0 the elevator moves upward, coming to rest again at time t = t 0. The downward acceleration of gravity is g. The acceleration of the elevator during this period is shown graphically above and is given analytically by a y (t) =! " 2! t 0 t. (1) a) Find the maximum speed of the elevator. b) Find the total distance traveled by the elevator. Problem 2: Traffic Violation A car is driving through a green light at t = 0 located at x = 0 with an initial speed v c,0 = 12 m!s -1. The acceleration of the car as a function of time is given by #% a c = 0; 0 < t < t = 1s 1 $. &%!(6 m " s -3 )(t! t 1 ); 1 s < t < t 2 a) Find the speed and position of the car as a function of time. b) Graph the speed and position of the car as a function of time. c) A bicycle rider is riding at a constant speed of v b,0 and at t = 0 is 17 m behind the car. The bicyclist reaches the car when the car just comes to rest. Find the speed of the bicycle.

3 Problem 3 Catching a bus At the instant a traffic light turns green, a car starts from rest with a given constant acceleration,5.0! 10 "1 m #s -2. Just as the light turns green, a bus, traveling with a given constant speed, 1.6! 10 1 m "s -1, passes the car. The car speeds up and passes the bus some time later. How far down the road has the car traveled, when the car passes the bus? Problem 4 Softball Catch A softball is hit over a third baseman s head. The third baseman, as soon as the ball is hit, turns around and runs straight backwards at a constant speed of 7.0m!s "1 for a time interval 2.0s and catches the ball at the same height it left the bat. The third baseman was initially 18 m from home plate. What is the initial speed and angle of the softball when it left the bat? Ignore air resistance. Problem 5: Jumping off a cliff A person, standing on a vertical cliff a height h above a lake, wants to jump into the lake but notices a rock just at the surface level with its furthest edge a distance s from the shore. The person realizes that with a running start it will be possible to just clear the rock, so the person steps back from the edge a distance d and starting from rest, runs at an acceleration that varies in time according to ax = b1t and then leaves the cliff horizontally. The person just clears the rock. Find s in terms of the given quantities d, b, h, and the gravitational constant g. You may neglect all air resistance. 1

4 Problem 6 Accelerating Wedge A 45 o wedge is pushed along a table with constant acceleration A! (see figure for choice of axes and positive directions) according to an observer at rest with respect to the table. A block of mass m slides without friction down the wedge. Find its acceleration with respect to an observer at rest with respect to the table. Write down a plan for finding the magnitude of the acceleration of the block. Make sure you clearly state which concepts you plan to use to calculate any relevant physical quantities. Also clearly state any assumptions you make. Be sure you include any free body force diagrams or sketches that you plan to use. Problem 7 Pulleys and Ropes Constraint Conditions Consider the arrangement of pulleys and blocks shown in the figure. The pulleys are assumed massless and frictionless and the connecting strings are massless and unstretchable. Denote the respective masses of the blocks as m 1, m 2 and m 3. The upper pulley in the figure is free to rotate but its center of mass does not move. Both pulleys have the same radius R. a) How are the accelerations of the objects related? b) Draw force diagrams on each moving object. c) Solve for the accelerations of the objects and the tensions in the ropes.

5 Problem 8 Block and Pulley A block of mass m 1 rests motionless on an inclined plane that makes an angle! with the horizontal. The coefficient of static friction between the block and plane is µ s. A massless inextensible string is attached to one end of the block, passes over a fixed pulley, around a second freely suspended pulley, and is finally attached to a fixed support. The left hand part of the string is parallel to the plane. The sections of the string coming off the suspended pulley are vertical. The pulleys are massless, but an object of mass m 2 is hung from the suspended pulley. Gravity acts downward. Part 1: Assume that if there were no friction the block would slide up the inclined plane. In what follows, express your answers in terms of m 1, m 2,!, µ s and the acceleration of gravity g (or some subset of these parameters). a) Draw separate free body diagrams for the system consisting of the object hanging from the suspended pulley plus the suspended pulley, and the block on the inclined plane, showing and labeling carefully, all the forces that act on the objects. b) Find the magnitude of the static friction force. c) What is the condition on the masses of the blocks such that the block on the incline plane just slides upward. Part 2: Now assume that the block on the incline plane is sliding upward. The coefficient of kinetic friction between the block and plane is µ k. In what follows, express your answers in terms of m 1, m 2,!, µ k and the acceleration of gravity g (or some subset of these parameters). d) Draw separate free body diagrams for the second (freely suspended) pulley, and the block on the inclined plane, showing and labeling carefully, all the forces that act on the objects. e) Find the magnitude of the acceleration of the block on the incline plane.

6 Problem 9: Winter Sports Calvin and Hobbes are riding on a sled. They are trying to jump the gap between two symmetrical ramps of snow separated by a distance W as shown above. Each ramp makes an angle! with the horizontal. They launch off the first ramp with a speed v L. Calvin, Hobbs and the sled have a total mass m. a) What value of the initial launch speed v L will result in the sled landing exactly at the peak of the second ramp? Express your answer in terms of some (or all) of the parameters m,!, W, and the acceleration of gravity g. Include in your answer a brief description of the strategy that you used and any diagrams or graphs that you have chosen for solving this problem. Make sure you clearly state which concepts you plan to use to calculate any relevant physical quantities. b) Hobbes was clutching a bag of tiger food when they left the first ramp. He got so excited while in the air that he let go of the bag at the top of the flight, lightening the total mass attached to the sled by 10%. Explain qualitatively how this will affect the results you found in a). c) Explain qualitatively what happens to the released bag of tiger food. d) There is friction between the sled and the snow which can be modeled by coefficient of kinetic friction µ k. What is the minimum value of µ k that will assure that the sled comes to rest somewhere on the second ramp? Assume an infinitely long ramp. Include in your answer a description of the plan that you will use to solve this part of the problem and any relevant diagrams or graphs that you find useful. Make sure you clearly state which concepts you plan to use to calculate any relevant physical quantities.

7 Problem 10 Bicycle and the Falling ipod A bicycle rider is traveling at a constant speed along a straight road and then gradually applies the brakes during a time interval 0 < t < t f until the bicycle comes to a stop. Assume that the magnitude of the braking force increases linearly in time according to F = bt, 0 < t < t f where b > 0 is a constant. The cyclist and bicycle have a total mass m. At the instant the person applies the brakes, a horizontal distance d from the rider, the wind blows and snaps an ipod off the branch of the tree with a horizontal speed v in the direction shown in the figure below. The ipod was initially a height h above the ground. The cyclist catches the ipod at the instant the cyclist has come to a stop. You may assume that the cyclist catches it at a height s above the ground. a) At what time t 1 did the cyclist catch the ipod? Express your answer in terms of the quantities, b, m, h, s, v po, d, and g as needed. b) What was the initial speed of the cyclist? You may leave your answer in terms of t from part a) and any other quantities as needed. 1 po

8 Problem 11 Two Block Pull Consider two blocks that are resting one on top of the other. The lower block has mass M 2 and is resting on a nearly frictionless surface. The upper block has mass M1 < M 2. Suppose the coefficient of static friction between the blocks is µ s. a) What is the maximum force with which the upper block can be pulled horizontally so that the two blocks move together without slipping? Draw as many free body force diagrams as necessary. Identify all action-reaction pairs of forces in this problem. b) What is the maximum force with which the lower block can be pulled horizontally so that the two blocks move together without slipping? Draw as many free body force diagrams as necessary. Identify all action-reaction pairs of forces in this problem.