1. Newton s Laws of Motion and their Applications Tutorial 1

Transcription

1 1. Newton s Laws of Motion and their Applications Tutorial On a planet far, far away, an astronaut picks up a rock. The rock has a mass of 5.00 kg, and on this particular planet its weight is 40.0 N. If the astronaut exerts an upward force of 46.2 N on the rock, what is its acceleration? (1.2 m/s 2 in the y direction) 1.2 A 92-kg water skier floating in a lake is pulled from rest to a speed of 12 m/s in a distance of 25 m. What is the net force exerted on the skier, assuming his acceleration is constant? (260N) 1.3 A 42.0-kg parachutist lands moving straight downward with a speed of 3.85 m/s. (a) If the parachutist comes to rest with constant acceleration over a distance of m, what force does the ground exert on her? (b) If the parachutist comes to rest over a shorter distance, is the force exerted by the ground greater than, less than, or the same as in part (a)? Explain. (415N, y direction) 1.4 A 747 airliner lands and begins to slow to a stop as it moves along the runway. If its mass is kg, its speed is 27.0 m/s, and the net braking force is N, (a) what is its speed 7.50 s later? (b) How far has it traveled in this time? (17.8 m/s, 168m) 1.5 You hold a brick at rest in your hand. (a) How many forces act on the brick? (b) Identify these forces. (c) Are these forces equal in magnitude and opposite in direction? (d) Are these forces an action-reaction pair? Explain. 1.6 On vacation, your 1300-kg car pulls a 540-kg trailer away from a stoplight with an 2 acceleration of m s. (a) What is the net force exerted by the car on the trailer? (b) What force does the trailer exert on the car? (c) What is the net force acting on the car? (1.0kN x direction, -1.0kN x direction, 2.5kN x direction) 1.7 A force of magnitude 7.50 N pushes three boxes with masses m kg, m 1 = = kg, and m 3 = kg, as shown in Figure 1.7. Find the magnitude of the contact force (a) between boxes 1 and 2, and (b) between boxes 2 and 3. (6.64N, 3.91N) Figure Two boxes sit side-by-side on a smooth horizontal surface. The lighter box has a mass of 5.2 kg, the heavier box has a mass of 7.4 kg. (a) Find the contact force between these boxes when a horizontal force of 5.0 N is applied to the light box. (b) If the 5.0-N force is applied to the heavy box instead, is the contact force between the boxes the same as, greater than, or less than the contact force in part (a)? Explain. (c) Verify your answer to part (b) by calculating the contact force in this case. (3.0N, 2.1N) 1.9 A farm tractor tows a 3900-kg trailer up a 16 incline with a steady speed of 3.0 m/s. What force does the tractor exert on the trailer? (Ignore friction.) (11kN) 1. Newton s Laws of Motion and their Applications 1 of 8

2 1.10 A shopper pushes a 7.5-kg shopping trolley up a 13 incline, as shown in Figure Find the magnitude of the horizontal force, F, r 2 needed to give the trolley an acceleration of 1.41 m s. (28N) Figure Two crewmen pull a raft through a lock, as shown in Figure One crewman pulls with a force of 130 N at an angle of 34 relative to the forward direction of the raft. The second crewman, on the opposite side of the lock, pulls at an angle of 45. With what force should the second crewman pull so that the net force of the two crewmen is in the forward direction? (0.10kN) Figure To give a 19-kg child a ride, two teenagers pull on a 3.7-kg sled with ropes, as indicated in Figure Both teenagers pull with a force of 55 N at an angle of 35 relative to the forward direction, which is the direction of motion. In addition, the snow exerts a retarding force on the sled that points opposite to the direction of motion, and has a magnitude of 57 N. Find the acceleration of the sled and child. (1.5m/s 2, x direction) Figure Newton s Laws of Motion and their Applications 2 of 8

3 1.13 A 65-kg skier speeds down a trail, as shown in Figure The surface is smooth and inclined at an angle of 22 with the horizontal. (a) Find the direction and magnitude of the net force acting on the skier. (b) Does the net force exerted on the skier increase, decrease, or stay the same as the slope becomes steeper? Explain. (0.24kN) Figure Figure 1.14 shows the Earth, Moon, and Sun (not to scale) in their relative positions at the time when the Moon is in its third quarter phase. Though few people realize it, the force exerted on the Moon by the Sun is actually greater than the force exerted on the Moon by the Earth. In fact, the force exerted on the Moon by the Sun has a magnitude of F SM = N, whereas the force exerted by the Earth has a magnitude of only F EM = N. These forces are indicated to scale in Figure Find (a) the direction and (b) the magnitude of the net force acting on the Moon. (c) Given that the mass of the Moon is M M = kg, find the magnitude of its acceleration at the time of the third-quarter phase. (24.5 above x axis, N, m/s 2 ) Figure Suppose a rocket launches with an acceleration of 30 2 an 88-kg astronaut aboard this rocket? (3.5kN). m/s 2. What is the apparent weight of 1.16 As part of a physics experiment, you stand on a bathroom scale in an elevator. Though your normal weight is 610 N, the scale at the moment reads 730 N. (a) Is the acceleration of the elevator upward, downward, or zero? Explain. (b) Calculate the magnitude of the elevator s acceleration. (c) What, if anything, can you say about the velocity of the elevator? Explain. (1.9m/s 2 ) 1.17 When you lift a bowling ball with a force of 82 N, the ball accelerates upward with an acceleration a. If you lift with a force of 92 N, the ball s acceleration is 2a. Find (a) the weight of the bowling ball, and (b) the acceleration a. (72N, 1.4m/s 2 ) 1.18 A 23-kg suitcase is being pulled with constant speed by a handle that is at an angle of 25 above the horizontal. If the normal force exerted on the suitcase is 180 N, what is the force F applied to the handle? (0.11kN) 1. Newton s Laws of Motion and their Applications 3 of 8

4 1.19 A 9.3-kg child sits in a 3.7-kg high chair. (a) Draw a free-body diagram for the child, and find the normal force exerted by the chair on the child. (b) Draw a free-body diagram for the chair, and find the normal force exerted by the floor on the chair. (91N, 0.13kN) 1.20 A 5.0-kg bag of potatoes sits on the bottom of a stationary shopping cart. (a) Sketch a freebody diagram for the bag of potatoes. (b) Now suppose the cart moves with a constant velocity. How does this affect your free-body diagram? Explain An ant walks slowly away from the top of a bowling ball, as shown in Figure If the ant starts to slip when the normal force on its feet drops below one-half its weight, at what angle θ does slipping begin? (60 ) Figure A child goes down a playground slide with an acceleration of 116. m/s 2. Find the coefficient of kinetic friction between the child and the slide if the slide is inclined at an angle of 31.0 below the horizontal. (0.436) 1.23 When you push a 1.80-kg book resting on a tabletop, it takes 2.25 N to start the book sliding. Once it is sliding, however, it takes only 1.50 N to keep the book moving with constant speed. What are the coefficients of static and kinetic friction between the book and the tabletop? (0.127, ) 1.24 To move a large crate across a rough floor, you push on it with a force F at an angle of 21 below the horizontal, as shown in Figure Find the force necessary to start the crate moving, given that the mass of the crate is 32 kg and the coefficient of static friction between the crate and the floor is (0.25kN) Figure A person places a cup of coffee on the roof of her car while she dashes back into the house for a forgotten item. When she returns to the car, she hops in and takes off with the coffee cup still on the roof. (a) If the coefficient of static friction between the coffee cup and the roof of the car is 0.24, what is the maximum acceleration the car can have without causing the cup to slide? Ignore the effects of air resistance. (b) What is the smallest amount of time in which the person can accelerate the car from rest to 15 m/s and still keep the coffee cup on the roof? (2.4m/s 2, 6.4s) 1. Newton s Laws of Motion and their Applications 4 of 8

5 1.26 The coefficient of kinetic friction between the tires of your car and the roadway is µ. (a) If your initial speed is v and you lock your tires during braking, how far do you skid? Give your answer in terms of v, µ, and m, the mass of your car. (b) If you double your speed, what happens to the stopping distance? (c) What is the stopping distance for a truck with twice the mass of your car, assuming the same initial speed and coefficient of kinetic friction? (v 2 /2µg) 1.27 A 50.0-kg person takes a nap in a backyard hammock. Both ropes supporting the hammock are at an angle of 15.0 above the horizontal. Find the tension in the ropes. (948N) 1.28 A backpack full of books weighing 52.0 N rests on a table in a physics laboratory classroom. A spring with a force constant of 150 N/m is attached to the backpack and pulled horizontally, as indicated in Figure (a) If the spring is pulled until it stretches 2.00 cm and the pack remains at rest, what is the force of friction exerted on the backpack by the table? (b) Does your answer to part (a) change if the mass of the backpack is doubled? Explain. (3N) Figure The equilibrium length of a certain spring with a force constant of k = 250 N/m is 0.18 m. (a) What is the magnitude of the force that is required to hold this spring at twice its equilibrium length? (b) Is the magnitude of the force required to keep the spring compressed to half its equilibrium length greater than, less than, or equal to the force found in part (a)? Explain. (45N) 1.30 A picture hangs on the wall suspended by two strings, as shown in Figure The tension in string 1 is 1.7 N. (a) Is the tension in string 2 greater than, less than, or equal to 1.7 N? Explain. (b) Verify your answer to part (a) by calculating the tension in string 2. (c) What is the weight of the picture? (0.85N, 2.0N) Figure Newton s Laws of Motion and their Applications 5 of 8

6 1.31 The pulley system shown in Figure 1.31 is used to lift a 52-kg crate. Note that one chain connects the upper pulley to the ceiling and a second chain connects the lower pulley to the crate. Assuming the masses of the chains, pulleys, and ropes are negligible, determine (a) the force r F required to lift the crate with constant speed, (b) the tension in the upper chain, and (c) the tension in the lower chain. (0.26kN downwards, 0.51kN, 0.51kN) Figure In Figure 1.32 we see two blocks connected by a string and tied to a wall. The mass of the lower block is 1.0 kg; the mass of the upper block is 2.0 kg. Given that the angle of the incline is 31, find the tensions in (a) the string connecting the two blocks and (b) the string that is tied to the wall. (5.1N, 15N) Figure After a skiing accident, your leg is in a cast and supported in a traction device, as shown in Figure Find the magnitude of the force r F exerted by the leg on the small pulley. (By Newton s third law, the small pulley exerts an equal and opposite force on the leg.) Let the mass m be 2.50 kg. (42.5N) Figure Newton s Laws of Motion and their Applications 6 of 8

7 1.34 You want to nail a 1.6-kg board onto the wall of a barn. To position the board before nailing, you push it against the wall with a horizontal force r F to keep it from sliding to the ground. (Figure 1.34) (a) If the coefficient of static friction between the board and the wall is 0.79, what is the least force you can apply and still hold the board in place? (b) What happens to the force of static friction if you push against the wall with a force greater than that found in part (a)? (20N) Figure Two blocks are connected by a string, as shown in Figure The smooth inclined surface makes an angle of 35 with the horizontal, and the block on the incline has a mass of 5.7 kg. The mass of the hanging block is m = 3. 2 kg. Find (a) the direction and (b) the magnitude of the hanging block s acceleration. (-0.076m/s 2 ) Figure A 3.50-kg block on a smooth tabletop is attached by a string to a hanging block of mass 2.80 kg, as shown in Figure The blocks are released from rest and allowed to move freely. (a) Is the tension in the string greater than, less than, or equal to the weight of the hanging mass? Find (b) the acceleration of the blocks and (c) the tension in the string. (4.36m/s 2, 15.3N) Figure Newton s Laws of Motion and their Applications 7 of 8

8 1.37 Two buckets of sand hang from opposite ends of a rope that passes over an ideal pulley. One bucket is full and weighs 110 N; the other bucket is only partly filled and weighs 63 N. (a) Initially, you hold onto the lighter bucket to keep it from moving. What is the tension in the rope? (b) You release the lighter bucket and the heavier one descends. What is the tension in the rope now? (c) Eventually the heavier bucket lands and the two buckets come to rest. What is the tension in the rope now? (110N, 80N, 63N) 1.38 When you take your 1200-kg car out for a spin, you go around a corner of radius 57 m with a speed of 15 m/s. The coefficient of static friction between the car and the road is Assuming your car doesn t skid, what is the force exerted on it by static friction? (4.7kN) 1.39 Find the linear speed of the bottom of a test tube in a centrifuge if the centripetal acceleration there is 52,000 times the acceleration of gravity. The distance from the axis of rotation to the bottom of the test tube is 7.5 cm. (200m/s) 1.40 Driving in your car with a constant speed of 12 m/s, you encounter a bump in the road that has a circular cross-section, as indicated in Figure If the radius of curvature of the bump is 35 m, find the apparent weight of a 67-kg person in your car as you pass over the top of the bump. (380N) Figure Referring to Question 1.40, at what speed must you go over the bump if people in your car are to feel weightless? (19m/s) 1. Newton s Laws of Motion and their Applications 8 of 8