1) Rizo the Rat (mass m = 0.5 kg) is back! Now he is cut loose...

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1) Rizo the Rat (mass m = 0.5 kg) is back! Now he is cut loose and running in a wheel having radius R = 20 cm and mass M = 3 kg that is free to roll on the ground. (This time there are no spokes, just the wheel.) See Figure 18, left. (i) The wheel starts at rest. To accelerate it, Rizo runs up one side to an angle 0 = 45° as shown in the first diagram in Figure 16. Rizo maintains that angle by running and spinning up the wheel, causing it to move forward with acceleration a. First draw a diagram identifying all forces on the wheel. Don't forget the force of friction! (ii) Write down the equations of Newton's Second Law for both the linear x-direction motion and the rotational motion. For the axis of rotation, take the center of the wheel. Then solve for acceleration a. Express your answer in terms of m, M, 0, and R. Then substitute numbers: you should get a = 0.107 m/s². (iii) What is the relationship between the speed at which Rizo runs relative to the wheel and the speed of the wheel relative to the ground? Rizo can run at maximum speed of Umar = 3 m/s. How long does it take Rizo to attain maximum speed? You should find a time of 28.1 s. (iv) Once at maximum speed, Rizo abruptly stops running and grabs onto the wheel. What is the velocity of the wheel to right after Rizo grabs hold? Express your answer in terms of Umar, m, M, R, and 0. Then substitute numbers: you should get v0 = 2.57 m/s. (v) What is the total kinetic energy of Rizo and the wheel right after Rizo grabs the wheel? Express your answer in terms of m, M, R and vo. Then substitute numbers: you should obtain KE = 11.6 J. (vi) Rizo does not loosen his grip on the wheel. Is the wheel moving fast enough that it will rotate all the way around bringing Rizo over the top and back to his starting point? R a (0) Figure 18: Diagrams for Problems 1 (Left) and 2 (Right). 1) Rizo the Rat (mass m = 0.5 kg) is back! Now he is cut loose and running in a wheel having radius R = 20 cm and mass M = 3 kg that is free to roll on the ground. (This time there are no spokes, just the wheel.) See Figure 18, left. (i) The wheel starts at rest. To accelerate it, Rizo runs up one side to an angle 0 = 45° as shown in the first diagram in Figure 16. Rizo maintains that angle by running and spinning up the wheel, causing it to move forward with acceleration a. First draw a diagram identifying all forces on the wheel. Don't forget the force of friction! (ii) Write down the equations of Newton's Second Law for both the linear x-direction motion and the rotational motion. For the axis of rotation, take the center of the wheel. Then solve for acceleration a. Express your answer in terms of m, M, 0, and R. Then substitute numbers: you should get a = 0.107 m/s². (iii) What is the relationship between the speed at which Rizo runs relative to the wheel and the speed of the wheel relative to the ground? Rizo can run at maximum speed of Umar = 3 m/s. How long does it take Rizo to attain maximum speed? You should find a time of 28.1 s. (iv) Once at maximum speed, Rizo abruptly stops running and grabs onto the wheel. What is the velocity of the wheel to right after Rizo grabs hold? Express your answer in terms of Umar, m, M, R, and 0. Then substitute numbers: you should get v0 = 2.57 m/s. (v) What is the total kinetic energy of Rizo and the wheel right after Rizo grabs the wheel? Express your answer in terms of m, M, R and vo. Then substitute numbers: you should obtain KE = 11.6 J. (vi) Rizo does not loosen his grip on the wheel. Is the wheel moving fast enough that it will rotate all the way around bringing Rizo over the top and back to his starting point? R a (0) Figure 18: Diagrams for Problems 1 (Left) and 2 (Right).