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SP221/2141&4341 Chapter 7 Homework Due Monday October 24, 2016 Question 1: A conveyor belt moves horizontally at a fixed constant speed v. A box of mass m is placed onto the belt, initially at rest. The coefficient of kinetic friction between the box and the conveyor belt is µk . The box initially slips on the conveyor belt, but eventually it is also moving at a speed v and from then on does not slip against the conveyor belt. How much thermal energy is produced by friction between the box and conveyor belt as the box comes to the same speed as the conveyor belt? Give your answer in terms of the given quantities (v, m, µk ) and g. Hint: First determine the work done by the belt on the box, i.e. determine the change in the box’s kinetic energy. Then determine the work done by the box on the conveyor belt (this is the hard part of the problem). This might help: Suppose that the box travels a distance d before it reaches speed v, how far does the conveyor belt move over the same period of time? No net work is done on the belt (it maintains the same constant speed v) so the work done on the conveyor belt by its motor must have the same magnitude but opposite sign as the work done on the belt by the box. The difference between the work done by the motor and the change in the box’s kinetic energy must be lost to thermal energy. [5 points] Question 2: A mountain climber of mass m is at the top of a large hemisphere of frictionless ice. Assume that the radius of this ice block is R. Starting from rest at the top, the climber slides down the ice. At what angle θ does she lose contact with the ice? (Hints: Determine her speed as a function of θ. Use this to determine the normal force acting her as a function of θ.) [10 points] θ Question 3: (Tipler & Mosca Problem 7-70) A block of mass m rests on a plane inclined at an angle θ with the horizontal (Figure 7-51). A spring with force constant k is attached to the block. The coefficient of static friction between the block and plane is µs . The spring is pulled upward along the plane very slowly. (a) What is the extension of the spring the moment the block begins to move? (b) The block stops moving just as the extension of the contracting spring reaches zero. Express µk (the coefficient of kinetic friction) in terms of µs and θ. [15 points] SP221/2141&4341 Chapter 7 Homework Due Monday October 24, 2016 Question 4: Two blocks, of masses m1 =2.50 kg and m2 =5.00 kg are connected to a spring with spring constant k=1250 N/m that has one end fixed to the wall as shown in the figure. The horizontal surface is frictionless and the pulley is ideal. The blocks are released from rest with the spring relaxed (at its equilibrium position). [10 points] 2.5 kg 5.0 kg (a) What maximum distance does the hanging block fall before momentarily stopping? (b) What is the maximum speed that the block will have while falling? (Hint: when the blocks are at their maximum speed, the net force on either block is F~net = ~0.) x Question 5: A conservative force acts only in the x-direction. This force is given by Fx (x) = (x2 +4) 2, where Fx has units of newtons when x has units of meters. Suppose that the potential energy associated with this conservative force is defined to be U0 = 0 at x = 0. What is the potential energy function, U (x)? [10 points] PROGRAMMING PROBLEM Question 6: A crane motor does work at a constant power of 100 kW. The crane is used to lift a box of mass 300 kg. The box is initially at rest and on the ground at time t=0. The motor is then turned on, and the crane lifts the box upwards. Graph both the speed and height of the box as a function of time, between 0 s and 2.5 s. Also plot the box’s kinetic energy, the potential energy of the box-earth system, and the total mechanical energy of the box-earth system over the same time frame. [20 points]