A Grand EntranceYou are designing an apparatus to support an actor of mass 80.0 kg who is to "fly" down to the stage during the performance of a play. You attach the actor's harness to a 150 kg sandbag by means of a lightweight steel cablerunning smoothly over two frictionless pulleys as in figure (a). You need 3.75 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, thesandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us call the initial angle that the actor's cable makes with the vertical 0. What is the maximum value ở can have before the sandbaglifts off the floor?(a) An actor uses some dever staging to makes entrance.(b) The free-body diagram for the actor at the bottom of the circularpath. (c) The free-body diagram for the sandbag if the normal forcefrom the floor goes to zero.MacoragActorSandbagmactoraSOLUTIONConceptualize We must use several concepts to solve this problem. Imagine what happens as the actor approaches the bottom of the swing. At the bottom, the cable is ---Select-- v and must support his weight as well as providecentripetal acceleration of his body in the upward direction. At this point in his swing, the tension in the cable is the highest and the sandbag is most likely to lift off the floor.Categorize Looking first at the swinging of the actor from the initial point to the lowest point, we model the actor and the Earth as an isolated system. We ignore air resistance, so there are no ---Select---v forces acting. Youmight initially be tempted to model the system as nonisolated because of the interaction of the system with the cable, which is in the environment. The force applied to the actor by the cable, however, is always perpendicular to eachelement of the displacement of the actor and hence does no work. Therefore, in terms of energy transfers across the boundary, the system is isolated.Analyze We first find the actor's speed as he arrives at the floor as a function of the initial angle e and the radius R of the circular path through which he swings.From the isolated system model, make the appropriate reduction of the fully expanded conservation of energy equation for the ---Select---v system:AK + AU, -0Let y, be the initial height of the actor above the floor and v,, be his speed at the instant before he lands. (Notice that K, -0 because the actor starts from rest and that U,- O because we define the configuration of the actor at thefloor as having a gravitational potential energy of zero. Use the following as necessary: mactor Ve 9, R, and y. Do not substitute numerical values; use variables only.)-)-(--(1)From the geometry in figure (a), notice that y, - 0, so y,- R - R cos(0) - R(1 - cos(0)). Use this relationship in Equation (1) and solve for v. (Use the following as necessary: g, mactor and R. Do not substitute numerical values;use variables only.)(2) v - Categorize Next, focus on the instant the actor is at the lowest point. Because the tension in the cable is transferred as a force applied to the sandbag, we model the actor at this instant as a particle ---Select---Because the actor moves along a circular arc, he experiences at the bottom of the swing a centripetal acceleration of directed upward.Analyze Apply Newton's second law from the particle under a net force model to the actor at the bottom of his path, using the free-body diagram in figure (b) as a guide, and recognizing the acceleration as centripetal. (Use thefollowing as necessary: R, g, and v, Do not substitute numerical values; use variables only.)v?SF,-T- mactor9 - mactormactor(3) T-Categorize Finally, notice that the sandbag lifts off the floor when the upward force exerted on it by the cable ---Select--- v the gravitational force acting on it; the normal force from the floor is zero when that happens. We do not,however, want the sandbag to lift off the floor. The sandbag must remain at rest, so we model it as a particle in equilibrium.Analyze A force T of the magnitude given by Equation (3) is transmitted by the cable to the sandbag. If the sandbag remains at rest but is just ready to be lifted off the floor if any more force were applied by the cable, the normalforce on it becomes zero and the particle in equilibrium model tells us that T? v mpagg as in figure (c).Substitute this condition and Equation (2) into Equation (3):2gR(1 - cos(0))mpagg - mactor9g + mactor-Solve for cos(0) and substitute the given parameters, then solve for (in degrees) (Due to the nature of this problem, do not use rounded intermediate values in your calculations-including answers submitted in WebAssign.):3mactor- moag.2mactor1.cos(e) -Finalize Here we had to combine several analysis models from different areas of our study. Notice that the length R of the cable from the actor's harness to the leftmost pulley did not appear in the final algebraic equation for cos(0).Therefore, the final answer is independent of which of the following?O mactorORO mpagEXERCISEThe director and his staff forget it is the body-double stuntman who is flying, not the actor. The stuntman's mass is 14 kg more than the actor. At the last second, you are asked to calculate how much extra mass, AM (in kg), to addto the sandbag in order to maintain the same maximum angle(Due to the nature of this problem, do not use rounded intermediate values in your calculations-including answers submitted in WebAssign.)HintkgNeed Help? Read It

Question

A Grand EntranceYou are designing an apparatus to support an actor of mass 80.0 kg who is to &#34;fly&#34; down to the stage during the performance of a play. You attach the actor's harness to a 150 kg sandbag by means of a lightweight steel cablerunning smoothly over two frictionless pulleys as in figure (a). You need 3.75 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, thesandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us call the initial angle that the actor's cable makes with the vertical 0. What is the maximum value ở can have before the sandbaglifts off the floor?(a) An actor uses some dever staging to makes entrance.(b) The free-body diagram for the actor at the bottom of the circularpath. (c) The free-body diagram for the sandbag if the normal forcefrom the floor goes to zero.MacoragActorSandbagmactoraSOLUTIONConceptualize We must use several concepts to solve this problem. Imagine what happens as the actor approaches the bottom of the swing. At the bottom, the cable is ---Select-- v and must support his weight as well as providecentripetal acceleration of his body in the upward direction. At this point in his swing, the tension in the cable is the highest and the sandbag is most likely to lift off the floor.Categorize Looking first at the swinging of the actor from the initial point to the lowest point, we model the actor and the Earth as an isolated system. We ignore air resistance, so there are no ---Select---v forces acting. Youmight initially be tempted to model the system as nonisolated because of the interaction of the system with the cable, which is in the environment. The force applied to the actor by the cable, however, is always perpendicular to eachelement of the displacement of the actor and hence does no work. Therefore, in terms of energy transfers across the boundary, the system is isolated.Analyze We first find the actor's speed as he arrives at the floor as a function of the initial angle e and the radius R of the circular path through which he swings.From the isolated system model, make the appropriate reduction of the fully expanded conservation of energy equation for the ---Select---v system:AK + AU, -0Let y, be the initial height of the actor above the floor and v,, be his speed at the instant before he lands. (Notice that K, -0 because the actor starts from rest and that U,- O because we define the configuration of the actor at thefloor as having a gravitational potential energy of zero. Use the following as necessary: mactor Ve 9, R, and y. Do not substitute numerical values; use variables only.)-)-(--(1)From the geometry in figure (a), notice that y, - 0, so y,- R - R cos(0) - R(1 - cos(0)). Use this relationship in Equation (1) and solve for v. (Use the following as necessary: g, mactor and R. Do not substitute numerical values;use variables only.)(2) v - Categorize Next, focus on the instant the actor is at the lowest point. Because the tension in the cable is transferred as a force applied to the sandbag, we model the actor at this instant as a particle ---Select---Because the actor moves along a circular arc, he experiences at the bottom of the swing a centripetal acceleration of directed upward.Analyze Apply Newton's second law from the particle under a net force model to the actor at the bottom of his path, using the free-body diagram in figure (b) as a guide, and recognizing the acceleration as centripetal. (Use thefollowing as necessary: R, g, and v, Do not substitute numerical values; use variables only.)v?SF,-T- mactor9 - mactormactor(3) T-Categorize Finally, notice that the sandbag lifts off the floor when the upward force exerted on it by the cable ---Select--- v the gravitational force acting on it; the normal force from the floor is zero when that happens. We do not,however, want the sandbag to lift off the floor. The sandbag must remain at rest, so we model it as a particle in equilibrium.Analyze A force T of the magnitude given by Equation (3) is transmitted by the cable to the sandbag. If the sandbag remains at rest but is just ready to be lifted off the floor if any more force were applied by the cable, the normalforce on it becomes zero and the particle in equilibrium model tells us that T? v mpagg as in figure (c).Substitute this condition and Equation (2) into Equation (3):2gR(1 - cos(0))mpagg - mactor9g + mactor-Solve for cos(0) and substitute the given parameters, then solve for (in degrees) (Due to the nature of this problem, do not use rounded intermediate values in your calculations-including answers submitted in WebAssign.):3mactor- moag.2mactor1.cos(e) -Finalize Here we had to combine several analysis models from different areas of our study. Notice that the length R of the cable from the actor's harness to the leftmost pulley did not appear in the final algebraic equation for cos(0).Therefore, the final answer is independent of which of the following?O mactorORO mpagEXERCISEThe director and his staff forget it is the body-double stuntman who is flying, not the actor. The stuntman's mass is 14 kg more than the actor. At the last second, you are asked to calculate how much extra mass, AM (in kg), to addto the sandbag in order to maintain the same maximum angle(Due to the nature of this problem, do not use rounded intermediate values in your calculations-including answers submitted in WebAssign.)HintkgNeed Help? Read It

Accepted Answer

Given:
Mass of the actor:
MA= 80 kg
Mass of the bag
MB= 150 kg
Distance between two pulley = 3.75 m