Chapter 8- AP physics



Does Conservation have the Potential to save Energy?What is potential energy?PotentialEnergy definitionEquationUnitVector?You drop a 2 kg book t o a friend who stands on the ground at a distance of 10 m below.How much work does the gravitational force do on the book as it drops to your friend’s hands?What is the change in the gravitational potential energy of the book-Earth system?How is potential energy related to gravitational force? m1m2A system consists of two objects having masses m1 and m2 (m1<m2). The objects are connected by a massless string, hung over a pulley as shown above, and then released. When the object of mass m2 has descended a distance h, the potential energy of the system has decreased by(m2-m1)ghm2gh(m1+m2)gh? (m1+m2)ghHow is potential energy related to force?The graph above shows potential energy U versus position x for an object. Which of the following graphs shows the force F exerted on the object as a function of position x?A 10 kilogram body is constrained to move along the x axis. The potential energy U of the body in joules is given as a function of its position x in meters byU(x)=6x2-4x+3The force on the particle at x=3 meters is32 N in +x direction32 N in -x direction45 N in +x direction45 N in –x direction98 N in +x directionFind the work/potential energy for a Hooke’s Law spring. What about other springs?Consider a spring with a restoring force given by F(x) =-bx3+cx5. If the potential energy is chosen to be zero at the equilibrium position, which of the following functions gives the elastic potential energy as a function of the spring’s displacement?U(x)=5cx4-3bx2U(x)=3bx2-5cx4U(x)=(cx6/b)-(bx4/4)U(x)=(bx4/4)-(cx6/6)None of the above2105025108585A specially designed spring is stretched from equilibrium to the distances x given in the table above, and the restoring force F is measured and recorded in each case. What is the potential energy of the spring when it is stretched 3 m from equilibrium?1257300539752276475149225 In an experiment, a student wishes to use a spring to accelerate a cart along a horizontal, level track. The spring is attached to the left end of the track, as shown above, and produces a nonlinear restoring force of magnitude Fs=As2+Bs, where s is the distance the spring is compressed, in meters. A measuring tape, marked in centimeters, is attached to the side of the track. The student places five photogates on the track at the locations shown. Derive an expression for the potential energy U as a function of the compression s. Express your answer in terms of A, B, s, and fundamental constants.In a preliminary experiment, the student pushes the cart of mass 0.3 kg into the spring, compressing the spring 0.04m. For this spring A=200 N/m2 and B = 150 N/m. The cart is release from rest. Calculate the speed of the cart immediately after it loses contact with the springConservation of Mechanical EnergyWhat is a nonconservative force? If one isn’t present then what is true? What is a conservative force? Why doesn’t friction have a potential energy?Think Projectile on the Moon!BCAConsidering the horizontal motion only, graph potential and kinetic energy for the football (projectile) as a function of heightConsidering the vertical motion only, graph potential and kinetic energy for the football (projectile) as a function of height A ball is thrown upward. At a height of 10 meters above the ground, the ball has a potential energy of 50 joules (with the potential energy equal to zero at the ground level) and is moving upward with a kinetic energy of 50 joules. Air friction is negligible. The maximum height reached by the ball is most nearly10m20m30m40m50mA block initially at rest is allowed to slide down a frictionless ramp and attains a speed v at the bottom. To achieve a speed of 2v at the bottom, how many times as high must a new ramp be?123456386461067945An adult exerts a horizontal force on a swing that is suspended by a rope of length L, holding it at an angle θ with the vertical. The child in the swing has a weight W and dimensions that are negligible compared to L. The adult releases the swing from rest In terms of W and θ determine the tension in the rope as the swing passes through its lowest point.3107055210820A ball of mass m, suspended by a cord of length L is displaced along its arc until it is elevated a vertical distance of L/2 above its lowest position at point P. The ball is then released from rest and swings like a pendulum The speed of the ball as it swings through point P is. gL2gLmgLmgL2mgL-85852097790A particle of mass m slides down a fixed, frictionless sphere of radius R. starting from rest at the top.a. In terms of m, g, R. and θ, determine each of the following for the particle while it is sliding on the sphere.The kinetic energy of the particleThe centripetal acceleration of the mass38646105056505In the figure to the right, the pulley has negligible mass, and both it and the inclined plane are frictionless. Block A has a mass of 1.0 kg, block B has a mass of 2.0 kg, and angle 30o. If the blocks are released from rest with the connecting cord taut, what is their total kinetic energy when block B has fallen 25 cm? You are to perform an experiment investigating the conservation of mechanical energy involving a transformation from initial gravitational potential energy to translational kinetic energy.(a) You are given the equipment listed below, all the supports required to hold the equipment, and a lab table.On the list below, indicate each piece of equipment you would use by checking the line next to each item.____ Track ____ Meterstick ____ Set of objects of different masses____ Cart ____ Electronic balance ____ Lightweight low-friction pulley____ String ____ Stopwatch(b) Outline a procedure for performing the experiment. Include a diagram of your experimental setup. Label the equipment in your diagram. Also include a description of the measurements you would make and a symbol for each measurement.(c) Give a detailed account of the calculations of gravitational potential energy and translational kinetic energy both before and after the transformation, in terms of the quantities measured in part (b).(d) After your first trial, your calculations show that the energy increased during the experiment. Assuming you made no mathematical errors, give a reasonable explanation for this result.(e) On all other trials, your calculations show that the energy decreased during the experiment. Assuming you made no mathematical errors, give a reasonable physical explanation for the fact that the average energy you determined decreased. Include references to conservative and nonconservative forces, as appropriate16129099060-1333501498600A constant 40 N/m is held horizontal over a horizontal air track, with one end attached to the air track. A light string is attached to the other end of the spring and connects it to a glider of mass m . The glider is pulled to stretch the spring an amount x from equilibrium and then released. Before reaching the photogate, the glider attains its maximum speed and the string becomes slack. The photogate measures the time t that it takes the small block on top of the glider to pass through. Information about the distance x and the speed v of the glider as it passes through the photogate are given below.Assuming no energy is lost, write the equation for conservation of mechanical energy that would apply to this situation.(b) On the grid below, plot v2 versus x2. Label the axes. including units and scale.47053587630(c)Draw a best-fit straight line through the data.Use the best-fit line to obtain the mass m of the glider.1371600445135(d) The track is now tilted at an angle θ as shown below. When the spring is unstretched, the center of the glider is a height h above the photogate. The experiment is repeated with a variety of values of x.Assuming no energy is lost, write the new equation for conservation of mechanical energy that would apply to this situation.Will the graph of v2 versus x2 for this new experiment be a straight line?_____ Yes_____NoJustify your answer.A spring loaded toy dart gun is used to shoot a dart straight up in the air and the dart reaches a maximum height of 24 m. The same dart is shot straight up a second time from the same gun, but this time the spring is compressed only half as far before firing. How far does the dart go this time, neglecting friction and assuming and ideal spring?96 m48 m24 m12 m6 m3 mTwo students are trying to hit a piece of paper on the floor with a plastic ball fired from a projectile launcher that is mounted on a table. The paper is 2.2 m horizontally from the edge of the table. Arlo compresses the spring 1.10 cm, but the center of the plastic ball falls 27.0 cm short of the center of the paper. How far should Alice compress the spring to hit the paper? Assume that neither the spring nor the ball encounters friction in the projectile launcher.027305In the system of two blocks and a spring shown above, blocks 1 and 2 are connected by a string that passes over a pulley. The initially unstretched spring connects block 1 to a rigid wall. Block 1 is released from rest, initially slides to the right, and is eventually brought to rest by the spring and by friction on the horizontal surface.Which of the following is true of the energy of the system during this process?The total mechanical energy of the system is conserved.The total mechanical energy of the system increases.The energy lost to friction is equal to the gain in the potential energy of the spring.The potential energy lost by block 2 is less in magnitude than the potential energy gained by the spring.The potential energy lost by block 2 is greater in magnitude than the potential energy gained by the spring.A 2.0 kg bread box on a frictionless 40 degree incline is connected by a cord that runs over a pulley, to a light spring of spring constant k=120 N/m. The box is released from rest when the spring is unstretched. Assume that the pulley is massless and frictionless.What is the speed of the box when it has moved 10 cm down the incline?How far down the incline from its point of release does the box slide before momentarily stoppingWhat are the magnitude and direction of the acceleration of the box at the instant it momentarily stops?A 20 kg mass, released from rest, slides 6 meters down a frictionless plane inclined at an angle of 30? with the horizontal and strikes a spring of spring constant k = 200 newtons/meter as shown in the diagram above. Assume that the spring is ideal, that the mass of the spring is negligible, and that mechanical energy is conserved. Determine the speed of the block just before it hits the spring. Determine the distance the spring has been compressed when the block comes to rest. Is the speed of the block a maximum at the instant the block strikes the spring? Justify your answer. Potential Energy Curves have Potential and Kinetic energies!On a graph of potential energy as a function of position what is your area?What does a point tell you?What does nonconservative tell you?What is the slope of a potential energy vs. displacement graph?What does it mean to be in equilibrium? Where are the equilibrium points on a potential energy versus displacement graph?A particle is released from rest at point A as shown. If no nonconservative forces are present, which of the following statements is true?UArGDFBECThe particle oscillates between points B and FThe particle cannot reach point GThe particle moves in simple harmonic motionThe particle experiences no force at points C,D, and EThe particle’s speed is maximized at point EU(x)1.0Jx0Jx3x2x1x0 -1.0J -2.0JA conservative force has the potential energy function U(x), shown by the graph above. A particle moving in one dimension under the influence of this force has kinetic energy 1.0 joule when it is at position x1. Which of the following is a correct statement about the motion of the particle?a. It oscillates with maximum position x2 and minimum position xo.b. It moves to the right of x3 and doe not return.c. It moves to the left of x0and does not returnd. It comes to rest at either x0 or x2e. It cannot reach either x0 or x2U(r)3U0Uor2roroIf the particle is released from rest at position r0, its speed at position 2r0 is most nearlya. b. c. d. e. If the potential energy function is given by U(r)=br-3/2+c, where b and c are constants, which of the following is an expression for the force on the particle?a. b. c. d. 2br-1/2+cre. A particle of mass m moves in a conservative force field described by the potential energy function U(r) = a(r/b + b/r), where a and b are positive constants and r is the distance from the origin. The graph of U(r) has the following shape.7112074295a. In terms of the constants a and b, determine the following.The position ro at which the potential energy is a minimumThe minimum potential energy Uo,b. Sketch the net force on the particle as a function of r on the graph below, considering a force directed away from the origin to be positive, and a force directed toward the origin to be negative.-59055348615.The particle is released from rest at r=ro /2In terms of Uo, and m, determine the speed of the particle when it is at r = roA 3.0 kg object is moving along the x-axis in a region where its potential energy as a function of x is given as U(x) = 4.0x2 , where U is in joules and x is in meters. When the object passes the point x = -0.50 m, its velocity is +2.0 m/s. All forces acting on the object are conservative.Calculate the total mechanical energy of the object.Calculate the x-coordinate of any points at which the object has zero kinetic energy.Calculate the magnitude of the acceleration of the object as it passes x = 0.60 m.What if there is friction?The figure below shows 10 situations in which a block of mass m with initial speed vi is brought to a stop. For each situation, write your conservation of energy equation. y=0??y=0 Into a region of frictional force f, where it stops in distance d(b) Up a frictionless ramp at angle ? until it (momentarily) stops at height hy=0?y=0(d) Across a frictionless floor and against a spring of spring constant k, stopping (momentarily) when the spring is compressed by distance d(c) Up a ramp at angle ? against a frictional force f until it stops at height hy=0y=0(f) Downward by height h onto a lower floor and into a region of frictional force f, stopping in that region in distance dAgainst a spring of spring constant k and simultaneously into a region of frictional force f, stopping when the spring is compressed by distance dy=0y=0(h) Upward by height h onto a higher floor and then into a region of frictional force f, stopping in distance d in that region(g) Downward by height h onto a lower floor and then simultaneously against a spring of spring constant k and into a region of frictional force f, stopping when the spring is compressed by distance dy=0y=0(j) Upward by height h onto a higher floor and then simultaneously against a spring of spring constant k and into a region of frictional force f, stopping when the spring is compressed by distance d.(i) Upward by height h onto a higher floor and then against a spring of spring constant k, stopping when the spring is compressed by distance d; A rubber sphere of mass m is dropped from a cliff. As the sphere falls, it is subject to air drag. The drag force on the sphere has the magnitude bv2, where b is a constant drag coefficient and v is the instantaneous speed of the sphere. Determine the energy dissipated by the drag force during the fall if the sphere is released at height h and reaches its terminal speed before hitting the ground in terms of the given quantities and fundamental constants. From the top of a 70 meter high building, a 1 kilogram ball is thrown directly downward with an initial speed of 10 meters per second. If the ball reaches the ground with a speed of 30 meters per second, the energy lost to friction is most nearly0J100 J300 J400 J700 J A stone is launched upward into the air. In addition to the force of gravity, the stone is subject to a frictional force due to air resistance. The time the stone takes to reach the top of its flight path is______ The time it takes to return from the top to its original position.Larger thanEqual toSmaller thanA 5.0 kg block is projected at 5.0 m/s up a plane that is inclined at 30 degrees with the horizontal. How far up along the plane does the block goa. if the plane is frictionless if the coefficient of kinetic friction between the block and the plane is 0.40?In the latter case, what is the increase in thermal energy of block and plane during the block’s ascent?If the block then slides back down against the frictional force, what is the block’s speed when it reaches the original projection point? A 1400 kg block of granite is pulled up an incline at a constant speed of 1.34 m/s by a cable and winch. (The incline is 40 m horizontally and 30 m vertically) The coefficient of kinetic friction between the block and the incline is 0.40. What is the power due to the force applied to the block by the cable? ................
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