For a situation involving gravity near earth s surface, a = g = jg. Show. that for that case v 2 = v 0 2 g(y y 0 ).

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1 Reading: Energy 1, 2. Key concepts: Scalar products, work, kinetic energy, work-energy theore; potential energy, total energy, conservation of echanical energy, equilibriu and turning points. 1.! In 1-D with constant acceleration we had the forula v 2 = v a(x x 0 ). It was noted that this only applies in 1-D. Take the 3-D equation for v for constant acceleration and show that the general forula is v 2 = v a (r r 0 ). For a situation involving gravity near earth s surface, a = g = jg. Show that for that case v 2 = v 0 2 g(y y 0 ). 2.! Today one can easure the acceleration of gravity g by dropping an object and aking a video of its fall. But in Galileo s tie the fall was too rapid to ake accurate easureents, so he used an incline to reduce the acceleration. Shown are (a) today s experient, and (b) Galileo s. In both the object starts fro rest. In case (b) the object slides down a frictionless incline. Neglect air resistance. h h (a) (b) θ! Coent on the validity of each stateent and explain your reasoning. The work done by gravity is the sae in both cases. There is no work done by any force other than gravity in either case. The object will reach the floor with the sae speed in both cases. The power input to the object is the sae in both cases. e.! If the easured acceleration in (b) is a, then g = a/sinθ.

2 3.! A particle of ass is acted on by a constant net force. It starts fro rest at t = 0. Show that the power input to the particle is proportional to t. Show that the kinetic energy of the particle is proportional to t 2. Show that the power input is proportional to x, where x is the distance traveled since the start. Show that the distance traveled during any tie interval is proportional to the increase in v 2 during that interval, i.e., Δx Δv 2. 4.! A block of ass is being lifted at constant speed. The pulleys are frictionless and assless and the rope is assless. A person pulls on the left end of the rope with tension T. When the person s hand oves down distance d, through what distance does the block ove up? Ans: d/2. What is the upward force exerted on the block by the rope, in ters of T? Ans: 2T. What is T in ters of and g? Ans: g/2 T Copare the power input by the person with the power delivered to the block. Ans: Sae. 5.! A ball of ass attached to a string of length R is rotating clockwise in a vertical circle as shown. When it is at the top of the circle at A the tension in the string is zero. Give answers in ters of, R and g. What is the ball s speed at that point? Ans: Rg. A C How uch work is done by gravity between A and B (the botto of the circle)? Ans: 2gR. B What is the ball s speed at B? Ans: 5Rg. e.! What is the ball s velocity (agnitude and direction) at C where the string is horizontal? Ans: 3Rg, vertically downward. What is the string tension at that point? Ans: 3g.

3 6.! A block of ass is attached to a spring of stiffness k, suspended fro a ceiling as shown. Originally it is at rest with the spring unstretched. It is released and falls. When it has fallen a distance y 1 it is oentarily at rest again. How uch work was done by gravity during the fall, in ters of, g and y 1? Ans: gy 1. Calculate the work done by the spring fro the definition of work, in ters of y 1 and k. Ans: 1 2 ky 1 2. Use the work energy theore to find y 1. Ans: 2g/k. What is the total force on the block (agnitude and direction) at this instant, in ters of and g? Ans: g, vertically upward. 7.! A girl, sitting on a seat in a bus oving with speed V, throws a ball of ass toward the front of the bus. As seen in the bus the ball is given initial speed v. What is the increase in kinetic energy of the ball as seen in the bus? Ans: 1 2 v2. As seen by an observer standing on the curb, what is the increase in kinetic energy of the ball? [What are the speeds of the ball before and after she throws it, as the observer sees the?] Ans: 1 2 (v2 + 2vV). Explain the difference. What work is done as seen by the observer on the curb, but not by the girl in the bus? 8.! Soe stateents about the law of conservation of echanical energy. Coent on their validity as written and give your reasons. The law says that the su of the kinetic and potential energies of a syste reains constant.! The law says that if the net external force is zero then the su of the kinetic and potential energies of a syste reains constant. The law says that if only conservative forces do work then the su of the kinetic and potential energies of a syste reains constant. The law says that if all the external forces are conservative then the su of the kinetic and potential energies of a syste reains constant.

4 9.! Soe short questions. Prove by direct calculation fro the definition that the work done by a constant force on a particle is independent of the path followed by the particle. The static friction force exerted on an object by a stationary surface does no work on the object. Why not? When you start your car oving fro rest there is an obvious increase in kinetic energy. But the external force (static friction) that causes the acceleration does no work. Does this contradict conservation of energy? Explain. The kinetic energy of a oving body is a property of its otion, regardless of what is happening to any other things it ight be interacting with. Explain the difference between that and the work done on the body. What about potential energy? 10.! A block of ass slides clockwise on a frictionless track in the for of a vertical circle. As the block oves over the top, what is the axiu speed it can have and stay on the track? Ans: Rg. θ Suppose when it is as shown it has speed v. What is the noral force exerted by the track at that instant, in ters of v, R, g and θ? [Draw a free body diagra with axes radial and tangent to the track.] Ans: gcosθ v 2 /R. Let the block go over the top with a non-zero but very sall speed. At what value of θ will be on the verge of leaving the track? [Find the speed in ters of θ.] Ans: cosθ = 2/3.

5 11.! Two identical blocks of ass are connected by a string and arranged as shown. The table is frictionless. The hanging block is attached to the spring, which is uncopressed when the syste is released fro rest. Let the coordinate y be easured downward fro the initial position of the hanging block, and let its gravitational potential energy be zero at that level. y Through what distance y 1 does the hanging block drop before its acceleration becoes zero? [The string is taut until then.] Ans: g/k. At that instant, what is its speed v 1? Ans: 2k g. After that the string becoes slack because the hanging block is slowing down while the one on the table is not. What value y 2 does the hanging block reach before coing oentarily to rest? [Assue the block on the table continues at constant speed.] Ans: g k ! In our later discussion of satellite orbits we will deal with a potential energy function which has the for!! U(r) = A r 2 B r! where A and B are positive constants and r > 0. Let the total energy be E. The turning points are the roots of the equation U = E. Any quadratic equation can be written as (r r 1 )(r r 2 ) = 0, where r 1,2 are the roots. Solve for r 1 + r 2 and r 1 r 2 in ters of the constants. Ans: r 1 + r 2 = B/E, r 1 r 2 = A/E. Take A = 1, B = 3 and E = 2. Plot U and E and show the turning points.!

15 Newton s Laws #2: Kinds of Forces, Creating Free Body Diagrams

15 Newton s Laws #2: Kinds of Forces, Creating Free Body Diagrams Chapter 15 ewton s Laws #2: inds of s, Creating ree Body Diagras 15 ewton s Laws #2: inds of s, Creating ree Body Diagras re is no force of otion acting on an object. Once you have the force or forces