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Name: _______________________ Mr. Kepple Work Problems Work and Energy β HW#1 Date: ___________ Period: _____ 1. How much work is done by the gravitational force when a 280-kg pile driver falls 2.80 m? 2. A 75.0-kg firefighter climbs a flight of stairs 20.0 m high. How much work is required? ππΊ = πΉπΊ π cos π πclimb = πΉclimb π cos π π = (ππ)π cos π π = (ππ)π cos π π = (280)(9.8)(2.80) cos(0) π = (75)(9.8)(20.0) cos(0) π = 7683.2 J π = 14700 J π β 7.7 × 103 J π = 1.47 × 104 J 3. What is the minimum work needed to push a 950-kg car 310 m up along a 9.0° incline? Ignore friction. Make sure you draw a free body diagram! +π¦ π +π₯ πΉπ The minimum work is when the car has constant velocity. Ξ£πΉ = 0 π ππ πΉπ β ππ sin π = 0 πΉπ = ππ sin π ππ = πΉπ π cos 0° ππ = (ππ sin π) π ππ = (950)(9.80)(310) sin 9.0° ππ = 451485.51 J ππ β 4.5 × 105 J 4. A box of mass 6.0 kg is accelerated from rest by a force across a floor at a rate of 2.0 m/s² for 7.0 s. Find the net work done on the box. 1 π = π£π π‘ + ππ‘ 2 2 1 π = 0 + ππ‘ 2 2 1 π = ππ‘ 2 2 π = πΉπ cos π 1 π = (ππ)( ππ‘ 2 ) cos 0° 2 1 π = ππ2 π‘ 2 2 1 π = (6.0)(2.0)2 (7.0)2 2 π = 588 J π β 590 J 5. A 17,000-kg jet takes off from an aircraft carrier via a catapult. The gases thrust out from the jetβs engines exert a constant force of 130 kN on the jet; the force exerted on the jet by the catapult is plotted in the figure. Determine: (a) The work done on the jet by the gases expelled by its engines during launch of the jet; and (b) the work done on the jet by the catapult during launch of the jet. (a) Work done by gases (b) Work done by catapult π = πΉπ cos π 1 π = (1100 × 103 + 65 × 103 )(85) 2 π = (130 × 103 )(85) cos 0° π = 49512500 J β 5.0 × 107 J π = 11050000 β 1.1 × 107 J 6. A 2200-N crate rests on the floor. How much work is required to move it at constant speed (a) 4.0 m along the floor against a drag force of 230 N, and (b) 4.0 m vertically? (a) Along the floor π πΉπ ππ = πΉπ π cos π π = (230)(4.0)(1) ππ = πππ cos 0° π = 920 J ππ = (2200)(4.0)(1) ππ π ππ ππ = 8800 J 7. At the top of a pole vault, an athlete pushes the pole with a force given by ( ) = 150 β 190 2 acting over a distance of 0.20 m. How much work is done on the athlete? 0.2 π= πΉπ ππ = πΉπ π cos π π π (b) Vertically (150π₯ β 190π₯ πΉππ₯ = 2 )ππ₯ 0 π = 75π₯ β 2 π = 75(0.2)2 β 190 3 8. Calculate the amount of work done by a force = 2 β which acts on an object during its journey along the axis from = 0.0 to = 1.0 m. π= 2 βπ₯ 1.0 ππ₯ = 2 0 0.2 3 π₯ 0 190 (0.2)3 β 0 β 0 3 π = 2.4933 β 2.5 π½ π= 2π₯1 2 (1 2) 1.0 0 π = 4(1 β 0) π = 4.0 J 1 β π₯ 2 ππ₯ Name: _______________________ Mr. Kepple Work-Energy Theorem Work and Energy β HW#2 Date: ___________ Period: _____ 1. The figure shows three forces applied to a trunk that moves leftward by 3.00 m over a frictionless floor. The force magnitudes are N, N, N, and the indicated angle is °. During the displacement what is the change in kinetic energy of the trunk? π πΉ π cos π ( )( ) cos ° π πΉ π cos π ( )( ) cos 2 ° β π πΉ π cos π ( )( ) cos J βπΎ J ° βπΎ J πnet π +π +π βπΎ β J 2. (a) How much work is done by the horizontal force N on the 18-kg block in the picture when the force pushes the block 5.0 m up along the 32° frictionless incline? (b) How much work is done by the gravitational force on the block during this displacement? (c) How much work is done by the normal force? (d) What is the speed of the block (assume it was zero initially) after this displacement? (a) Work done by πΉπ (b) Work by gravity ππ ππΊ ππ πΉπ π cos π ( ππ πΉπΊ π cos π 2 )( ) cos 2° ππΊ πππ cos π Jβ 4 J ππΊ ( 8)( 8)( ) cos 22° π£ ππΊ β4 7 4 J β β47 J (c) Work by normal force ππ (d) Speed of the block πΉπ π cos π(π£ β π£0 ) 2π π π£ 2( π β 4 7 4) ( 8) 4 82 m/s β 4 m/s ° 3. At an accident scene on a level road, investigators measure a carβs skid mark to be 98 m long. It was a rainy day and the coefficient of friction was estimated to be 0.38. Use these data to determine the speed of the car when the driver slammed on (and locked) the brakes. π π βππππ π ππ π ππ cos π ππππ cos 8 ° π£ βπΎ π£ π(π£ β π£0 ) π( β π£0 ) π£ β ππ£0 2πππ 2( 8)( 8)( 8) 27 2 m/s β 27 m/s 4. A 46.0-kg crate, starting from rest, is pulled across a floor with a constant horizontal force of 225 N. For the first 11.0 m the floor is frictionless, and for the next 10.0 m the coefficient of friction is 0.20. What is the final speed of the crate after being pulled these 21.0 m? βπΎ π π(π£ β π£0 ) π(π£ β ) π£ π +π πΉπ + πΉπ β ππππ 2 πΉ(π + π ) β ππππ π π πΉπ cos ° πΉπ π πΉπ cos ° + ππ cos 8 ° π πΉπ β ππππ 2 (22 )(2 ) β ( 2)(4 )( 8)( (4 ) π£ 2 8 m/s β ) m/s 5. A mass is attached to a spring which is held stretched a distance by a force , and then released. The spring compresses, pulling the mass. Determine the speed of the mass when the spring returns: (a) to its normal length ( ); (b) to half its original extension ( /2). [Ignore friction] (a) Speed at π₯ βπΎ 2 (b) Speed at π₯ ππ π(π£ β π£0 ) ππ£ + βπΎ 2 ππ₯ 2 π(π£ β π£0 ) (ππ₯)ππ₯ π₯/ πΉ π₯ π₯ πΉπ₯ π ππ π₯ 2 π£ π₯/2 ππ£ π₯ π 2 ππ£ π (π₯) β πΉ π₯ π₯ ππ£ π£ π₯ π₯/ π₯ 2 4 β 4 4 πΉπ₯ 4π Name: _______________________ Mr. Kepple Chapter 7: Questions Work and Energy β HW#3 Date: ___________ Period: _____ 1. The figure shows two horizontal forces that act on a block that is sliding to the right across a frictionless floor. The graph shows three plots of the blockβs kinetic energy versus time . Which of the plots best corresponds to the following situations? Justify each response. (a) (b) (c) Graph 2. The forces on the block are balanced, which means no acceleration occurs. As a result, the kinetic energy is constant. Graph 3. The net force points opposite to the direction of motion. The object will eventually come to rest and have zero kinetic energy. Graph 1. The net force points parallel to the direction of motion. The object will gain speed and the kinetic energy will increase. 2. The figure shows four graphs (draw to the same scale) of the component of a variable force (directed along an axis) versus the position of a particle on which the force acts. Rank the graphs according to the work done by the force on the particle from to , from most positive work first to most negative work last. Rank: b, a, c, d. The work done is the area under the curve. Graph (b) is the only graph with a net positive area. Graph (a) is next since it has a net area of zero. Finally graph (c) and then (d) since graph (c) has less negative area than graph (d). 3. Spring is stiffer than spring springs are compressedβ¦ . The spring force of which spring does more work if the (a) the same distance? Justify. (b) by the same applied force? Justify. Spring A. In order to compress each spring by an equal length, more force must be applied to the spring with the greater stiffness constant. As a result, the stiffer spring (spring A) does more work. Spring B. When both springs are compressed by an equal applied force, the spring with the smaller stiffness constant will be compressed a greater distance and as a result more work will be done. 4. In the figure, a greased pig has a choice of three frictionless slides along which to slide to the ground. Rank the slides according to how much work the gravitational force does on the pig during the descent, greatest first. Justify your ranking. All tie. Since gravity is a conservative force, the work done by gravity will depend only on the initial and final height of the pig. Along each of the three paths shown the pig will start and end at the same height. Therefore the work done by gravity is the same along all three paths. 5. In three situations, a briefly applied horizontal force changes the velocity of a hockey puck that slides over frictionless ice. The overhead views of the figure indicate, for each situation, the puckβs initial speed , its final speed , and the directions of the corresponding velocity vectors. Rank the situations according to the work done on the puck by the applied force, most positive first and most negative last. Justify your ranking. Ranking: c, b, a. According to the work-energy theorem, the work done on the object is equal to the change in its kinetic energy. Situation (c) is the only situation with positive work since the object gains speed. Situations (a) and (b) both have negative work since the puck loses speed. Kinetic energy is proportional to speed squared so more negative work is done in (a) since the magnitude of the speeds are greater than in (b). 6. The figure gives the component particle. If the particle begins at rest when it hasβ¦ (justify each response) of a force that can act on a , what is its coordinate (a) its greatest kinetic energy? 3 m. The gain in kinetic energy will be equal to the work done, in other words the area under the curve. (b) its greatest speed? 3 m. Kinetic energy depends on speed. The location with the most kinetic energy also has the most speed. (c) zero speed? (d) What is the particleβs direction of travel after it reaches m? m and m. The object starts at rest and returns to rest when the net work done is equal to zero. The area under the curve from 0 m to 6 m is zero. Negative π₯. The object returns to rest at 6 m and past this point is acted on by a negative net force. It will accelerate and gain speed in the negative π₯ direction. Potential Energy Mr. Kepple Name: _______________________ Work and Energy β HW#4 Date: ___________ Period: _____ 1. A particular spring obeys the force law β . (a) Is this force conservative? Explain. (b) Determine the form of the potential energy function. Yes this force is conservative. Since the force is a function of position, the work done by the force depends only on the initial and final position (the end points) of the object. Therefore, by definition, the force is conservative. π πΉ ππ₯ π ( ππ₯ π 1 ππ₯ 1 ππ₯ ππ₯ ) ππ₯ ππ₯ 1 5 ππ₯ 5 πΆ 2. The potential energy of an object is given by ( ) (a) What is the force, β ? (b) What force acts on the object at πΉ(π₯) ππ(π₯) ππ₯ πΉ(π₯) 6π₯ πΉ( ) 6( ) πΉ(π₯) π ( π₯ ππ₯ πΉ( ) 8N πΉ(π₯) (6π₯ π₯ m? ) ) 3. A particle constrained to move in one dimension is acted on by the force ( ) where is a constant with units appropriate to the SI system. Find the potential energy function ( ), if is arbitrarily defined to be zero at m, so that ( ) . π πΉ ππ₯ π π π₯ ππ₯ π₯ β ππ₯ π( ) π π π₯ πΆ π ( ) π π π₯ π 8 πΆ π π πΆ π 8 J 4. The figure shows a plot of potential energy versus position of a 2.0-kg particle that can travel only along an axis. Assume nonconservative forces are not involved. Suppose the particle is released at speed of 7.0 m/s, headed in the negative J with an initial direction. J Remember the work-energy theorem! Since and this means that . In other words, when an object gains potential energy it loses kinetic energy. (a) Does the particle have enough kinetic energy to reach m? π( ) π π( ) π J 1 πΎ (b) What is the magnitude and direction of the force acting on the particle as it begins to move to the left of m? ππ£ 1 πΉ ( )(7) ππ ππ₯ 9J πΉ Yes the particle can reach π₯ . It needs at least 30 J of kinetic energy and it has 49 J. 5. A spring has a spring constant 35.0 J of potential energy? ππ 1 πΉ N to the right of 82.0 N/m. How much must this spring be compressed to store ππ₯ ππ π π₯ π π₯ ( 5) 8 9 π₯β 9 6 9 6. A 1.60-m tall person lifts a 1.95-kg book off the ground so it is 2.20 m above the ground. What is the potential energy of the book relative to (a) the ground, and (b) the top of the personβs head? 6 book (a) relative to the ground (b) relative to the head head π ππβ π ππβ π ( 95)(9 8)( π ( 95)(9 8)( π 66 J ground π πβ J J ) πβ 5J 6)