HW#29 Ch 10: Rotation
Instructions: Be sure to write formulas and show substitutions to earn full credit.
1. A thin disk of radius R and mass M has a variable area mass density given by \(\sigma \left( r \right) = {\sigma _0}\left( {4 - \dfrac{{2r}}{R}} \right)\), where σ0 is a constant expressed in kg/m2 and r is the distance from the center of the disk in meters. If the disk’s moment of inertia has the form I = cMR2, find the value of the constant c and express it as a decimal.
2. Four dejected offensive linemen from the Cincinatti Bengals are sitting on an aluminum bench with a mass of 34.5 kg and a length of 4.57 m, having two leg supports located 0.914 m from each end. Suppose that each lineman has a mass of 143 kg (average mass of the Bengals offensive line), and that they are positioned 1 m, 1.5 m, 2m and 2.5m from one end of the bench, respectively. What are the forces that each leg provides if the bench is in static and rotational equilibrium?
3. Find the torque in unit vector notation if the lever arm is described by \(\vec r = - \hat i + 2\hat j + 2\hat k\), where \(\vec r \) is in meters, and the force is described by \(\vec F = - 30\hat i + 10\hat j - 20\hat k\), with \(\vec F \) measured in Newtons. You must show all steps on paper.
4. A grinding wheel that is initially at rest is mounted on an axis that is not frictionless. A motor applies a constant torque of 50 N∙m to the wheel for 20 s, giving the wheel an angular velocity of 600 rev/min. The motor is then shut off, and the wheel comes to rest 120 s later. Find (a) the moment of inertia of the wheel, and (b) the frictional torque, which is assumed to be constant both during the acceleration and deceleration phases.