21. A child of mass m stands on a platform which rotates freely at angular speed...

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21. A child of mass m stands on a platform which rotates freely at angular speed w. The child's centre of mass (COM) is initially at a radius of ro from the axis of the platform as shown in the figure below. The child now slowly pulls itself closer to the centre of the platform. COM m To (a) State the minimal force F(r) needed to pull the child closer to the centre as a function of the radius r of the child's COM. (b) Find a mathematical expression for the work done by the child when it has reached the centre of the platform. Hint: You may take the work done along a path to be W = JF(s)ds where F is the force in the direction of motion and s is the distance. If your expression contains an integral you do not have to perform the integration yet. (c) Into which form of energy is the child's work converted and what will qualitatively happen to the angular speed was the child approaches the centre? (d) The angular momentum J of the system of child and platform is a conserved quantity during the above process. It is defined to be J = Iw where I is the moment of inertia of the system which is given by Imr2 + Ip where Ip is the constant moment of interial of the platform. Find an expression for the angular speed w(r) as a function of the radius r of the child's COM. (e) Find the total work done by the child when it has reached r = 0. Hint: (a+ry2dr = 2(a+r²) [1] [1] [2] [2] [4] 21. A child of mass m stands on a platform which rotates freely at angular speed w. The child's centre of mass (COM) is initially at a radius of ro from the axis of the platform as shown in the figure below. The child now slowly pulls itself closer to the centre of the platform. COM m To (a) State the minimal force F(r) needed to pull the child closer to the centre as a function of the radius r of the child's COM. (b) Find a mathematical expression for the work done by the child when it has reached the centre of the platform. Hint: You may take the work done along a path to be W = JF(s)ds where F is the force in the direction of motion and s is the distance. If your expression contains an integral you do not have to perform the integration yet. (c) Into which form of energy is the child's work converted and what will qualitatively happen to the angular speed was the child approaches the centre? (d) The angular momentum J of the system of child and platform is a conserved quantity during the above process. It is defined to be J = Iw where I is the moment of inertia of the system which is given by Imr2 + Ip where Ip is the constant moment of interial of the platform. Find an expression for the angular speed w(r) as a function of the radius r of the child's COM. (e) Find the total work done by the child when it has reached r = 0. Hint: (a+ry2dr = 2(a+r²) [1] [1] [2] [2] [4]

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Given platform rotates freely with a angular speed w child slowly pull itself closer to center of the platform a the minimal force needed to pull the child closer to the center of the platform as a fu... View the full answer