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Date _________________ Period _________________ Name _________________________________ CHAPTER 5 Supplemental Problems DISPLACEMENT AND FORCE IN TWO DIMENSIONS 1. A small plane takes off and flies 12.0 km in a direction southeast of the airport. At this point, following the instructions of an air traffic controller, the plane turns 20.0 to the east of its original flight path and flies 21.0 km. What is the magnitude of the plane’s resultant displacement from the airport? 4. To get a cart to move, two farmers pull on ropes attached to the cart, as shown below. One farmer pulls with a force of 50.0 N in a direction 35.0 east of north, while the other exerts a force of 30.0 N in a direction 25.0 west of north. What are the magnitude and the direction of the combined force exerted on the cart? 2. A hammer slides down a roof that makes a 32.0 angle with the horizontal. What are the magnitudes of the components of the hammer’s velocity at the edge of the roof if it is moving at a speed of 6.25 m/s? 5. Takashi trains for a race by rowing his canoe on a lake. He starts by rowing along a straight path. Then he turns and rows 260.0 m west. If he then finds he is located 360.0 m exactly north of his starting point, what was his displacement along the straight path? 3. A worker has to move a 17.0-kg crate along a flat floor in a warehouse. The coefficient of kinetic friction between the crate and the floor is 0.214. The worker pulls horizontally on a rope attached to the crate, with a 49.0-N force. What is the resultant acceleration of the crate? 6. Mira received a 235-N sled for her birthday. She takes the sled out to a flat field. When she pushes it with a 45.0-N horizontal force, it slides along at a constant speed. What is the coefficient of kinetic friction between the coefficient of kinetic friction between the sled and the field? Chapter 5 Displacement and Force in Two Dimensions 1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Name _____________________________ Supplemental Problems 7. A rod supports a 2.35-kg lamp, as shown below. continued 9. A child shoves a small toboggan weighing 100.0 N up a hill, giving the toboggan an initial speed of 6.0 m/s. If the hill is inclined at an angle of 32 above the horizontal, how far along the hill will the toboggan slide? Assume that the coefficient of sliding friction between the toboggan and the hill is 0.15. a. What is the magnitude of the tension in the rod? b. Calculate the components of the force that the bracket exerts on the rod. 10. Two objects are connected by a string passing over a frictionless, massless pulley. As shown below, the block is on an inclined plane and the ball is hanging over the top edge of the plane. The block has a mass of 60.0 kg, and the coefficient of kinetic friction between the block and the inclined plane is 0.22. If the block moves at a constant speed down the incline, and the ball rises at a constant speed, what is the mass of the hanging ball? 8. A 25.0-kg crate has an adjustable handle so that it can be pushed or pulled by the handle at various angles. Determine the acceleration of the crate for each situation shown in the diagram, given that the coefficient of sliding friction between the floor and the bottom of the crate is 0.20. Chapter 5 Displacement and Force in Two Dimensions 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Supplemental Problems Teacher Support DISPLACEMENT AND FORCE IN TWO DIMENSIONS 1. A small plane takes off and flies 12.0 km in a direction southeast of the airport. At this point, following the instructions of an air traffic controller, the plane turns 20.0 to the east of its original flight path and flies 21.0 km. What is the magnitude of the plane’s resultant displacement from the airport? R2 A2 B2 2 AB cos R (12.0 km)2 (21.0 km)2 2(12.0 km)(21.0 km)(cos 160.0) 32.5 km 2. A hammer slides down a roof that makes a 32.0 angle with the horizontal. What are the magnitudes of the components of the hammer’s velocity at the edge of the roof if it is moving at a speed of 6.25 m/s? Fourth quadrant: vx . 0 and vy 0. vx v cos (6.25 m/s)(cos 32.0) 5.30 m/s vy v sin (6.25 m/s)(sin 32.0) 3.31 m/s Chapter 5 Displacement and Force in Two Dimensions 3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Supplemental Problems Teacher Support 3. continued A worker has to move a 17.0-kg crate along a flat floor in a warehouse. The coefficient of kinetic friction between the crate and the floor is 0.214. The worker pulls horizontally on a rope attached to the crate, with a 49.0-N force. What is the resultant acceleration of the crate? y-direction: FN Fg mg x-direction: Fnet, x Fp Ff max ma Ff k FN k mg ma Fp k mg a Fp k mg m 49.0 N (0.214)(17.0 kg)(9.8 N/kg) 17.0 kg 2 0.785 m/s 4. To get a cart to move, two farmers pull on ropes attached to the cart, as shown below. One farmer pulls with a force of 50.0 N in a direction 35.0° east of north, while the other exerts a force of 30.0 N in a direction 25.0° west of north. What are the magnitude and the direction of the combined force exerted on the cart? Rx Ax Bx A cos 1 B cos 2 Ry Ay By A sin 1 B sin 2 R Rx2 Ry2 ( A cos 1 B cos 2 )2 ( A sin 1 B sin 2 )2 ((50.0 N) (cos 55.0) (30.0 N)(cos 1150))2 ((50.0 N) (sin 550) (30.0 N)(sin 115.0))2 70.0 N R A sin 1 B sin2 tan1 y tan1 Rx A cos 1 B cos2 (50.0 N)(sin 55.0) (30.0 N)(sin 115.0) tan1 (50.0 N)(cos 55.0) (30.0 N)(cos 115.0) 76.8 R 70.0 N at 76.8north of east Chapter 5 Displacement and Force in Two Dimensions 4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Supplemental Problems Teacher Support 5. continued a 0 (since v constant) Takashi trains for a race by rowing his canoe on a lake. He starts by rowing along a straight path. Then he turns and rows 260.0 m west. If he then finds he is located 360.0 m exactly north of his starting point, what was his displacement along the straight path? Fp Ff Fp Ff k FN k Fg k Fp Fg 45.0 N 235 N 0.191 7. A rod supports a 2.35-kg lamp, as shown below. Rx 0.0 m, Ry 360.0 m Bx 260.0 m, By 0.0 m R A B, A R B Ax Rx Bx 0.0 m (260.0 m) a. What is the magnitude of the tension in the rod? y - direction : Ty Fg 260.0 m Ay Ry By 360.0 m 0.0 m 360.0 m A Ax2 Ay2 T sin mg (260.0 m)2 (360.0 m)2 mg sin (2.35 Kg)(9.8 N/kg) sin 25.0 54 N T 444.1 m Ay Ax 360.0 m tan1 54.16 260.0 m A 444.1 m at 54.16 north of east tan1 6. b. Calculate the components of the force that the bracket exerts on the rod. x - direction: Tx Fx 0 Fx Tcos (54 N)(cos 25.0) Fx 49 N, inward y - direction: Mira received a 235-N sled for her birthday. She takes the sled out to a flat field. When she pushes it with a 45.0-N horizontal force, it slides along at a constant speed. What is the coefficient of kinetic friction between the sled and the field? Fy Ty y -direction: FN Fg mg (2.35 kg) (9.8 N/kg) Fy 23.0 N, upward x -direction: Fp Ff ma Chapter 5 Displacement and Force in Two Dimensions 5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Supplemental Problems Teacher Support 8. continued A 25.0-kg crate has an adjustable handle so that it can be pushed or pulled by the handle at various angles. Determine the acceleration of the crate for each situation shown in the diagram, given that the coefficient of sliding friction between the floor and the bottom of the crate is 0.20. For (a) y -direction : FN Fg mg FN mg x -direction : Fnet, x Fp Ff Fnet, x max ma ma Fp Ff Fp F cos Ff k FN k mg Ff k FN k (mg F sin ) ma Fp k mg a Fp k mg Fp Fnet, x F cos k (mg F sin ) ma k g Fnet m F cos k (mg F sin ) m F cos k F sin ) k g m (70.0 N)(cos 30.0 ) (0.20)(70.0 N)(sin 30.0 ) 25.0 kg (0.20)(9.8 N/kg) a m m 70.0 N (0.20)(9.8 N/kg) 25.0 kg 0.84 m/s2 For (b) y -direction : FN Fg Fy FN mg F sin 0.18 m/s2 x -direction : Fnet, x Fp Ff Fnet, x max ma ma Fp Ff Chapter 5 Displacement and Force in Two Dimensions 6 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Supplemental Problems Teacher Support For (c) y -direction : FN Fg Fy FN mg F sin x -direction : Fnet, x Fp Ff Fnet, x max ma ma Fp Ff Fp F cos Ff k FN k (mg F sin ) Fnet, x F cos k (mg F sin ) ma Fnet m F cos k (mg F sin ) m F cos k F sin ) k g m (70.0 N)(cos 30.0) (0.20)(70.0 N)(sin 30.0) (0.20)(9.8 N/kg) 25.0 kg 0.74 m/s2 a Chapter 5 Displacement and Force in Two Dimensions 7 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. continued Supplemental Problems Teacher Support 9. continued A child shoves a small toboggan weighing 100.0 N up a hill, giving the toboggan an initial speed of 6.0 m/s. If the hill is inclined at an angle of 32° above the horizontal, how far along the hill will the toboggan slide? Assume that the coefficient of sliding friction between the toboggan and the hill is 0.15. y -direction : Fnet, y may 0 FN Fgy 0 FN Fgy mg cos x -direction : Fnet, x max ma Fgx mg sin Fgx Ff ma F Ff ma Ff k FN k mg cos mg sin k mg cos ma a g(sin k cos ) vf2 vi2 2a(xf xi ) 0 vi2 2ax d vi2 vi2 2a 2g(sin k cos ) (6.0 m/s)2 (2)(9.8 N/kg)(sin 32 (0.15) cos 32.0) 4.6 m, up the hill 10. Two objects are connected by a string passing over a frictionless, massless pulley. As shown below, the block is on an inclined plane and the ball is hanging over the top edge of the plane. The block has a mass of 60.0 kg, and the coefficient of kinetic friction between the block and the inclined plane is 0.22. If the block moves at a constant speed down the incline, and the ball rises at a constant speed, what is the mass of the hanging ball? Chapter 5 Displacement and Force in Two Dimensions 8 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Supplemental Problems Teacher Support v is constant, so there is no acceleration. Ball (m2 ) : y - direction : Fnet, y m2 ay 0 FT Fg 0 FT m2 g Block (m1 ): y - direction : (perpendicular to the incline) Fnet, y m1 ay 0 FN Fgy 0 FN Fgy m1 g cos Ff k FN k m1 g cos x - direction : (parallel to the incline) Fnet, X m1 aX 0 Fgx Ff FT 0 FgX m1 g sin m1g sin k m1g cos FT 0 m1g sin k m1g cos m2 g 0 m g sin k m1g cos m2 1 g m1 sin k m1 cos (60.0 kg)(sin 35.0) (0.22)(60.0 kg)(cos 35.0) 24 kg Chapter 5 Displacement and Force in Two Dimensions 9 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. continued