HW#30 Ch 10: Rotation
Instructions: Be sure to write formulas and show substitutions to earn full credit.
1. A motor provides a torque to a steel platter that varies with rotational speed, such that \(\tau \left( \omega \right) = {\tau _0}\left[ {1 - \dfrac{{\omega }}{{{\omega _0}}}} \right]\), where τ0 is the stall torque and ω0 is the rotational speed when there is no torque on the platter. (a) Write an expression for the power as a function of rotational speed. (b) Find the value of the rotational speed when the power is at a maximum. Hint: you will need to take a derivative. (c) Find the value of the torque when the power is at a maximum.
2. A wood screw is screwed into a piece of hard wood with a torque that varies with angle as \(\tau (\theta ) = \dfrac{{185\theta }}{{{\pi ^2}}} - \dfrac{{60}}{\pi }\) , where τ is in N·m and θ is in radians. What is the work required to turn the screw a full rotation, starting from θ = 0?
3. A pulsar is a rapidly rotating neutron star that emits regular pulses of electromagnetic radiation that can be detected from Earth. Chinese astronomers first discovered the Crab pulsar in the year 1054; it has a radius of 15 km and a mass of 4 x 1030 kg, which is twice as massive as our sun, although much smaller in size. The pulsar has a very fast rotation period of 33.4 milliseconds. (a) Calculate the velocity of a point on the equator as a fraction of the speed of light. (b) Determine the moment of inertia of the rotating pulsar if it is modeled as a uniform sphere. (c) The rotation period of the pulsar is actually slowing down at a rate of 20.4 ns per day. Calculate how much energy is dissipated each second as the pulsar slows down. (Write an expression for the rotational kinetic energy in terms of the period T, then take the derivative with respect to time.) (d) If the Sun radiates 4 x 1026 J of energy each second, how many times more energy does the Crab pulsar radiate each second?
4. Ben is in the library, where he sees a Bellerby & Co. antique globe, of diameter 80 cm and mass 4.5 kg, attached to a frictionless vertical spindle. In a frantic bid to turn back time, Ben wraps a string around the globe, passes it over a massless, frictionless pulley of radius 5 cm that is attached to the window, and ties the other end to a 3.5 kg brick. As the brick falls to the ground, it pulls on the globe, accelerating it around its vertical axis. If the globe is a uniform spherical shell (I = 2/3MR2) and the string is horizontal, calculate the magnitude of the brick’s acceleration.