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Problem Statement 1. A carousel with a radius of R = 3.0 meters is initially at rest. It is then given a constant angular acceleration α = 0.06 rad/s². A. Set-Up the Problem • Sketch the physical situation and label your sketch with appropriate quantities from the problem statement. Make a table of known/given information and unknown/wanted information (this may involve reading the rest of the problem before starting). List any physical assumptions you will be making to solve the problem. B. Solve (symbolically first, then plug in): • • What is the angular velocity of the carousel at t = 8.0 s? What is the tangential velocity of a child located at 2.5 meters from the center of the carousel at t = 8.0 s? What is the tangential acceleration of the child at that time? What is the child's radial acceleration at that time? Determine the magnitude of the child's total acceleration vector after 8.0 s. Add this vector to your diagram above and show the directions of the radial and tangential components. C. Sensemaking: Assess the validity of your work in the following ways: • Check the physical units of your expressions in part B. Compare the child's acceleration to g = 9.8 m/s/s? Is your answer for the total acceleration reasonable for a carousel? Units can be tricky, so I want you to think about the following as well: i. What are the units of angle? ii. What are the dimensions of angle? iii. What is the difference between units and dimensions? 2. A pulley of mass mp and radius R is attached to the edge of a table. A string of negligible mass is hung over the pulley and attached on one end to block 2 which hangs over the edge of the table. The other end is attached to block 1 which then slides along the table. The coefficient of kinetic friction between the table and block 1 is μk. At time t = 0 the blocks are released from rest. Assume mass 2 is large enough that the system accelerates. m m2 D. Set Up the Problem: • Establish an appropriate coordinate system(s) and draw a free body diagram for each block. Draw an extended free-body diagram for the pulley. An extended diagram, often called a torque diagram, shows: o The axis of rotation o the point of application of each force and their distance from the axis of rotation. • Identify any torques caused by those forces and the signs of the torques. Is the tension the same in each segment of the rope? Why or why not? E. Set Up and Solve Equations: Write the Newton's Second Law equations for m₁, Write the Newton's Second Law equation for m2, Write the Rotational Newton's Second Law equation for the pulley. How are the accelerations of the blocks related? How are they related to the angular acceleration of the pulley?