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☰ Search Explore Log in Create new account Upload × RECITATION WEEK 10 CHAPTER 7 7.27. A spring of force constant 300.0 N/m and unstretched length 0.240 m is stretched by two forces, pulling in opposite directions at opposite ends of the spring, that increase to 15.0 N. How long will the spring now be, and how much work was required to stretch it that distance? Set Up: Solve Fon spring kx for x to determine the length of stretch and use Won spring 12 kx 2 to assess the corresponding work. Solve: x Fon spring k 15.0 N 0.0500 m 300.0 N/m The new length will be 0.240 m 0.0500 m 0.290 m. The corresponding work done is, Won spring 1 300.0 N/m 0.0500 m 2 0.375 J 2 Reflect: Fon spring is always the force applied to one end of the spring, thus we did not need to double the 15.0 N force. Consider a free-body diagram of a spring at rest; forces of equal magnitude and opposite direction are always applied to both ends of every section of the spring examined. 7.31. How high can we jump? The maximum height a typical human can jump from a crouched start is about 60 cm. By how much does the gravitational potential energy increase for a 72 kg person in such a jump? Where does this energy come from? Solve: U grav mg( y f yi ) 72kg 9.80m / s 2 0.60m 420 J Reflect: This gravitational potential energy comes from elastic potential stored in his tensed muscles. 7.36. A certain spring stores 10.0 J of potential energy when it is stretched by 2.00 cm from its equilibrium position. (a) How much potential energy would the spring store if it were stretched an additional 2.00 cm? (b) How much potential energy would it store if it were compressed by 2.00 cm from its equilibrium position? (c) How far from the equilibrium position would you have to stretch the spring to store 20.0 J of potential energy? (d) What is the force constant of this spring? Set Up: Use U el 12 kx 2 for the potential energy. U 1 for x1 and U 2 for x2 are related by 2 U 2 / U1 x2 / x1 . x Solve: (a) U 2 U 1 2 x1 2 2 4.00 10 2 m 10.0 J 10.0 J 4.00 J 40.0 J 2 2 . 00 10 m (b) U el depends only on the magnitude of x, not on its sign; therefore, the elastic potential energy is U el 10.0 J. (c) x2 x1 (d) k U2 20.0 J 2.00 10 2 m 2.00 2.00 10 2 m 100cm / m 2.83cm U1 10.0 J 2U 210.0 J 2 x 2.00 10 2 m 2 5.00 10 4 N / m 7.45. Pebbles of weight w are launched from the edge of a vertical cliff of height h at speed v0. How fast (in terms of the quantities just given) will these pebbles be moving when they reach the ground if they are launched (a) straight up, (b) straight down, (c) horizontally away from the cliff, and (d) at an angle θ above the horizontal? (e) How would the answers to previous parts change if the pebbles weighed twice as much? 1 w 2 1 2 g v f and K i w g v0 . Let 2 2 1 w 2 v0 wh . 2 g Set Up:. Use K f U f K i U i with K f yi h and yf 0; this gives 1 w 2 v f 2 g Solve: (a)-(d) v f v0 2 gh (e) Since K and U are both proportional to the weight 2 w mg, w divides out of the expression and v f is unaffected by a change in weight. Reflect: The initial kinetic energy depends only on the initial speed and is independent of the direction of the initial velocity. 7.49. Two stones of different masses are thrown straight upward, one on earth and one on the moon (where gravity produces an acceleration of 1.67 m/s2). Both reach the same height. If the stone on the moon had an initial speed v, what was the initial speed of the stone on earth, in terms of v? Set Up: Apply K f U f K i U i to the motion of the stone on the moon with U i 0 and K f 0 at the maximum height. The conservation equation becomes U f K i with 1 vi v moon,i v and K i K moon,i mv 2 . Setting yf h and yi 0 gives 2 2 1 2 v mv mgh . Note that is the same on the earth and moon and let subscript 1 refer 2 g to the moon and subscript 2 refer to the earth. Solve: v1 / g1 v 2 / g 2 , v 2 v1 2 2 g2 9.80m / s 2 v1 2.42v1 g1 1.67m / s 2 Reflect: Gravity is stronger on earth so a larger initial speed is needed in order to achieve the same height as on the moon. 7.60. A 1.1 kg mass hangs from a spring of force constant 400.0 N/m. The mass is then pulled down 13.0 cm from equilibrium position and released. At the end of five complete cycles of vibration, the mass reaches only 10.0 cm from the equilibrium position. (a) How much mechanical energy is lost during these five cycles? (b) What percentage of mechanical energy is lost during the five cycles? Set Up: Applying Emech K U with K i K f 0, the change in mechanical energy is simply the difference between the initial and final elastic potential energy of the stretched spring: Emech,i Emech,f 12 k xi2 xf2 . Also define the initial position as the point of release and the final position as the lowest position after five complete cycles. Solve: (a) Emech,i Emech,f (b) The percentage lost is E 1 2 400.0 N/m 0.130 m 0.100 m 1.38 J. 2 mech ,i E mech, f E mech,i 2 1.38 J 100% 40.8% 3.38 J Reflect: Note that the spring, in its equilibrium position, is stretched by the weight of the hanging mass. Consequently, an alternative solution can be developed that explicitly incorporates gravitational potential energy. Let d be the distance the spring is stretched at equilibrium due to the weight of the object. F kd mg or 1.1 kg 9.80 m/s 2 mg d 2.70 3 102 m 0.027 m k 400 N/m Define a new reference point; let y yi 0 where the mass is 0.130 m below the equilibrium point. At this point, the spring is stretched di 0.130 m 0.027 m 0.157 m. The total potential energy is therefore: Emech,i U el U grav 12 kdi 2 mgyi 1 2 400 N/m 0.157 m 0 4.93 J 2 Then when the mass is 0.100 m below the equilibrium position, y yf 0.030 m and the spring is stretched df 0.100 m 0.027 m 0.127 m. The total energy is then: Emech,f U el U grav 12 kdf 2 mgyf 1 2 400 N/m 0.127 m 1.1 kg 9.80 m/s 0.030 m 3.55 J 2 2 The change in mechanical energy is thus 4.93 J 3.55 J 1.38 J, the same as found in the simplified calculation of part (a). Download 1. Science 2. 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