P8.14. m 1 > m 2. m 1 gh = 1 ( 2 m 1 + m 2 )v 2 + m 2 gh. 2( m 1. v = m 1 + m 2. 2 m 2v 2 Δh determined from. m 2 g Δh = 1 2 m 2v 2.

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Alyson Mathews
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. Two objects are connected by a light string passing over a light frictionless pulley as in Figure P8.3. The object of mass m is released from rest at height h.

1 . Two objects are connected by a light string passing over a light frictionless pulley as in Figure P8.3. The object of mass m is released from rest at height h. Using the principle of conservation of energy, (a) determine the speed of m just as m hits the ground. Find the maximum height to which m rises. P8.4 m > m (a) m gh = ( m + m )v + m gh v = ( m m )gh ( m + m ) Since m has kinetic energy, it will rise an additional height m v Δh determined from m g Δh = m v or from (a), Δh = v ( g = m m )h ( m + m ) m The total height m reaches is h + Δh = h. m + m. A light rigid rod is 77.0 cm long. Its top end is pivoted on a low- friction horizontal axle. The rod hangs straight down at rest, with a small massive ball attached to its bottom end. You strike the ball, suddenly giving it a horizontal

2 velocity so that it swings around in a full circle. What minimum speed at the bottom is required to make the ball go over the top of the circle? P8.5 The force of tension and subsequent force of compression in the rod do no work on the ball, since they are perpendicular to each step of displacement. Consider energy conservation of the ball-earth system between the instant just after you strike the ball and the instant when it reaches the top. The speed at the top is zero if you hit it just hard enough to get it there. initial final K i + U gi = K f + U gf mv i + 0 = 0 + mg v i = 4g = : ( 0.770) v i v i = 5.49 m s FIG. P A daredevil plans to bungee- jump from a balloon 65.0 m above a carnival midway. He will use a uniform elastic cord, tied to a harness around his body, to stop his fall at a point 0.0 m above the ground. Model his body as a particle, and the cord as having negligible mass and one that obeys Hooke s force law. In a preliminary test, hanging at rest from a m length of the cord, he finds that his body weight stretches it by.50 m. He will drop from rest at the point where the top end of a longer section of the cord is attached to the stationary balloon. (a) What length of cord should he use? What maximum acceleration will he experience? *P8.9 (a) For a 5-m cord the spring constant is described by F = kx, mg = k(.5 m). For a longer cord of length the stretch distance is longer so the spring constant is smaller in inverse proportion: here y i y f = 55 m = + x f k = 5 m mg = 3.33 mg.5 m ( K + U g + U s ) = K + U g + U s i f 0 + mgy i + 0 = 0 + mgy f + kx f mg( y i y f ) = kx f = mg 3.33 x f initial FIG. P8.9(a) final

3 55.0 m = 3.33 ( 55.0 m ) 55.0 m = m 83 m = = 0 = 38 ± only the value of less than 55 m is physical. = 38 ± mg k = m x max = x f = 55.0 m 5.8 m = 9. m F = ma +kx max mg = ma mg m mg = ma 5.8 m a =.77 g = 7. m s = 5.8 m

4. A particle of mass m = 5.00 kg is released from point A and slides on the frictionless track shown in Figure P8.4. Determine (a) the particle'ʹs speed at points B and C and the net work done by the gravitational force in moving the particle from A to C.

4 4. A particle of mass m = 5.00 kg is released from point A and slides on the frictionless track shown in Figure P8.4. Determine (a) the particle'ʹs speed at points B and C and the net work done by the gravitational force in moving the particle from A to C. P8.4 (a) ( ΔK) A B = W = W g = mgδh = mg mv B mv A = m( 9.80) (.80) v B = 5.94 m s A 5.00 m B 3.0 m C.00 m Similarly, v C = v A + g( ) = 7.67 m s FIG. P8.4 W g = mg 3.00 m A C = 47 J 5. A kg block and 00- kg block are connected by a string as in Figure P8.36. The pulley is frictionless and of negligible mass. The coefficient of kinetic friction between the 50- kg block and incline is Determine the change in the kinetic energy of the 50- kg block as it moves from A to B, a distance of 0.0 m

P8.36 F y = n mg cos37.0 = 0 n = mgcos 37.0 = 400 N f = µn = 0.50 400 N = 00 N fδx = ΔE mech ( 00) 0.0 = ΔU A + ΔU B + ΔK A + ΔK B ( 9.80) ( 0.0sin 37.0 ) = 5.

A uniform board of length is sliding along a smooth (frictionless) horizontal plane, as in Figure P8.39a. The board then slides across the boundary with a rough horizontal surface.

5 P8.36 F y = n mg cos37.0 = 0 n = mgcos 37.0 = 400 N f = µn = N = 00 N fδx = ΔE mech ( 00) 0.0 = ΔU A + ΔU B + ΔK A + ΔK B ( 9.80) ( 0.0sin 37.0 ) = ΔU A = m A g h f h i ΔU B = m B g h f h i = 50.0 = 00 ΔK A = m A v f v i ( 9.80) ( 0.0) = ΔK B = m B v ( f v i ) = m B ΔK m A = ΔK A A Adding and solving, ΔK A = 3.9 kj. FIG. P A uniform board of length is sliding along a smooth (frictionless) horizontal plane, as in Figure P8.39a. The board then slides across the boundary with a rough horizontal surface. The coefficient of kinetic friction between the board and the second surface is µ k. (a) Find the acceleration of the board at the moment its front end has traveled a distance x beyond the boundary. The board stops at the moment its back end reaches the boundary, as in Figure P8.39b. Find the initial speed v of the board. *P8.39 (a) et m be the mass of the whole board. The portion on the rough surface has mx mxg mass. The normal force supporting it is and the frictional force is µ k mgx = ma. Then a = µ k gx opposite to the motion. In an incremental bit of forward motion dx, the kinetic energy converted into internal energy is f k dx = µ k mgx dx. The whole energy converted is

6 mv = µ k mgx dx v = 0 µ k g = µ k mg x = µ k mg 0 7. A toy cannon uses a spring to project a g soft rubber ball. The spring is originally compressed by 5.00 cm and has a force constant of 8.00 N/m. When it is fired, the ball moves 5.0 cm through the horizontal barrel of the cannon, and there is a constant friction force of N between the barrel and the ball. (a) With what speed does the projectile leave the barrel of the cannon? At what point does the ball have maximum speed? (c) What is this maximum speed? P8.35 (a) ( K + U) i + ΔE mech = ( K + U) f : 0 + kx fδx = mv + 0 ( 8.00 N m) ( m) ( N) ( 0.50 m) = ( kg)v v = ( J) kg =.40 m s When the spring force just equals the friction force, the ball will stop speeding up. Here F s = kx ; the spring is compressed by N 8.00 N m = cm and the ball has moved 5.00 cm cm = 4.60 cm from the start. (c) Between start and maximum speed points, kx i fδx = mv + kx f v =.79 m s ( )( ) = 3 ( )v ( ) 8. The mass of a car is 500 kg. The shape of the body is such that its aerodynamic drag coefficient is D = and the frontal area is.50 m. Assuming that the drag force is proportional to v and neglecting other sources of friction, calculate the power required to maintain a speed of 00 km/h as the car climbs a long hill sloping at 3.0.

0 kg m3 Comparing the energy of the car at two points along the hill, = 38 N or where K i + U gi + ΔE = K f + U gf K i + U gi + ΔW e R( Δs) = K f + U gf ΔW e is the work input from the engine.

7 P8.50 v = 00 km h = 7.8 m s The retarding force due to air resistance is R = DρAv = (.50 m ) 7.8 m s.0 kg m3 Comparing the energy of the car at two points along the hill, = 38 N or where K i + U gi + ΔE = K f + U gf K i + U gi + ΔW e R( Δs) = K f + U gf ΔW e is the work input from the engine. Thus, ΔW e = R( Δs) + ( K f K i ) + ( U gf U gi ) Recognizing that K f = K i and dividing by the travel time Δt gives the required power input from the engine as # P = % $ ΔW e Δt & # ( = R Δs & # % ( + mg Δy & % ( = Rv + mgvsin θ ' $ Δt ' $ Δt ' + ( 500 kg) 9.80 m s sin 3.0 P = ( 38 N) 7.8 m s P = 33.4 kw = 44.8 hp 7.8 m s 9. A 0.0- kg block is released from point A in Figure P8.57. The track is frictionless except for the portion between points B and C, which has a length of 6.00 m. The block travels down the track, hits a spring of force constant 50 N/m, and compresses the spring m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between the block and the rough surface between B and C. P8.57 ΔE mech = fδx E f E i = f d BC kx mgh = µmgd BC µ = mgh kx mgd BC = 0.38 FIG. P8.57

8 0. Jane, whose mass is 50.0 kg, needs to swing across a river (having width D) filled with man- eating crocodiles to save Tarzan from danger. She must swing into a wind exerting constant horizontal force F, on a vine having length and initially making an angle θ with the vertical (Fig. P8.65). Taking D = 50.0 m, F = 0 N, = 40.0 m, and θ = 50.0, (a) with what minimum speed must Jane begin her swing in order to just make it to the other side? Once the rescue is complete, Tarzan and Jane must swing back across the river. With what minimum speed must they begin their swing? Assume that Tarzan has a mass of 80.0 kg. (a) From takeoff to alighting for the Jane-Earth system ( K + U g ) + W wind = K + U g i f mv i + mg cosθ 50 kg v i + 50 kg 9.8 m s + FD( ) = 0 + mg( cosφ) 40 mcos 50 ( 40 mcos8.9 ) = 50 kg 9.8 m s 0 N( 50 m) 50 kg v i J J = J v i = 947 J 50 kg = 6.5 m s For the swing back

9 . A single conservative force acting on a particle varies as, where A and B are constants and x is in meters. (a) Calculate the potential- energy function U(x) associated with this force, taking U = 0 at x = 0. Find the change in potential energy and the change in kinetic energy as the particle moves from x =.00 m to x = 3.00 m. P8.40 (a). A potential energy function for a two- dimensional force is of the form U = 3x 3 y 7x. Find the force that acts at the point (x, y). P8.4 mv i + mg cosφ 30 kg v i + 30 kg 9.8 m s + FD ( +) = 0 + mg( cosθ) 40 mcos8.9 ( 40 mcos 50 ) = 30 kg 9.8 m s + 0 N( 50 m) 30 kg v i J J = J v i = 30 kg J = 9.87 m s x U = ( Ax + Bx )dx = Ax Bx ΔU = A 3.00 Fdx = 3.00 m.00 m $ ΔK = & 5.00 A % 3 B ' ) ( F x = U x = 3x 3 y 7x x F y = U y = 3x 3 y 7x y [ (.00) ] = ( 9x y 7) = 7 9x y = ( 3x 3 0) = 3x 3 F = [ 3 (.00) 3 ] B F = F xˆ ( Ax+ Bx ) ˆi N = 5.00 A B Thus, the force acting at the point x, y is j = 7 9x y j. i + F yˆ ˆ i 3x 3ˆ

The negative root tells how high the mass will rebound if it is instantly glued to the spring. We want

The negative root tells how high the mass will rebound if it is instantly glued to the spring. We want 8.38 (a) The mass moves down distance.0 m + x. Choose y = 0 at its lower point. K i + U gi + U si + E = K f + U gf + U sf 0 + mgy i + 0 + 0 = 0 + 0 + kx (.50 kg)9.80 m/s (.0 m + x) = (30 N/m) x 0 = (60