Physics Homework Solutions

9.5 A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to 𝜃(𝑡) = 𝛾𝑡 + 𝛽𝑡 3 , where 𝛾 = 0.428 𝑟𝑎𝑑/𝑠 , 𝛽 = 0.0100 𝑟𝑎𝑑/𝑠 3 (a) Calculate the angular velocity of the merry-go-round as a function of time. (b) What is the initial value of the angular velocity? (c) (i) Calculate the instantaneous value of the angular velocity 𝜔𝑧 at 𝑡 = 4.55 𝑠. (c) (ii) Calculate the average velocity 𝜔𝑎𝑣−𝑧 for the time interval 𝑡 = 0 to 𝑡 = 4.55 𝑠 9.15 A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has mass 35.0 kg and diameter 74.0 cm. The power is off for 34.0 s, and during this time the flywheel slows due to friction in its axle bearings. During the time the power is off, the flywheel makes 170 complete revolutions. (a) At what rate is the flywheel spinning when the power comes back on? (b) (i) How long after the beginning of the power failure would it have taken the flywheel to stop if the power had not come back on? (b) (ii) How many revolutions would the wheel have made during this time? 9.25 An advertisement claims that a centrifuge takes up only 0.127 m of bench space but can produce a radial acceleration of 2600 g at 4330 rev/min. (a) Calculate the required radius of the centrifuge. (b) Is this claim realistic? 9.38 An airplane propeller is 1.73 m in length (from tip to tip) with mass 109 kg and is rotating at 2500 rpm (rev/min) about an axis through its center. You can model the propeller as a slender rod. (a) What is its rotational kinetic energy? (b) Suppose that, due to weight constraints, you had to reduce the propeller's mass to 75.0% of its original mass, but you still needed to keep the same size and kinetic energy. What would its angular speed have to be, in rpm? 9.54 A thin uniform rod of mass 𝑀 and length 𝐿 is bent at its center so that the two segments are now perpendicular to each other. (a) Find its moment of inertia about an axis perpendicular to its plane and passing through the point where the two segments meet. (b) Find its moment of inertia about an axis perpendicular to its plane and passing through the midpoint of the line connecting its two ends. 9.62 You are designing a rotating metal flywheel that will be used to store energy. The flywheel is to be a uniform disk with radius 25.0 cm. Starting from rest at t = 0, the flywheel rotates with constant angular acceleration 3.00 𝑟𝑎𝑑/𝑠 2 about an axis perpendicular to the flywheel at its center. (a) If the flywheel has a density (mass per unit volume) of 8600 𝑘𝑔/𝑚3 , what thickness must it have to store 800 J of kinetic energy at t = 8.00 s? 9.79 Two metal disks, one with radius 𝑅1 = 2.55 cm and mass 𝑀1 = 0.850 kg and the other with radius 𝑅2 = 5.02 cm and mass 𝑀2 = 1.58 kg, are welded together and mounted on a frictionless axis through their common center. (a) What is the total moment of inertia of the two disks? (b) A light string is wrapped around the edge of the smaller disk, and a 1.50 kg block is suspended from the free end of the string. If the block is released from rest at a distance of 2.03 m above the floor, what is its speed just before it strikes the floor? (c) Repeat part b, this time with the string wrapped around the edge of the larger disk. 9.72 A uniform, solid disk with mass m and radius R is pivoted about a horizontal axis through its center. A small object of the same mass m is glued to the rim of the disk. (a) If the disk is released from rest with the small object at the end of a horizontal radius, find the angular speed when the small object is directly below the axis. Express your answer in terms of some or all of the variables m, R, and the acceleration due to gravity g. 9.86 The Crab Nebula is a cloud of glowing gas about 10 light-years across, located about 6500 light years from the earth The figure shows the cloud of glowing gas in space. It is the remnant of a star that underwent a supernova explosion, seen on earth in 1054 a.d. Energy is released by the Crab Nebula at a rate of about 5.0 × 1031 W, about 105 times the rate at which the sun radiates energy. The Crab Nebula obtains its energy from the rotational kinetic energy of a rapidly spinning neutron star at its center. This object rotates once every 0.0331 s, and this period is increasing by 4.22 × 10−13 s for each second of time that elapses. (a) If the rate at which energy is lost by the neutron star is equal to the rate at which energy is released by the nebula, find the moment of inertia of the neutron star. Express your answer in kilogram meters squared. (b) Theories of supernovae predict that the neutron star in the Crab Nebula has a mass about 1.4 times that of the sun. Modeling the neutron star as a solid uniform sphere, calculate its radius in kilometers. (c) (i) What is the linear speed of a point on the equator of the neutron star? (c) (ii) Compare the speed in part (c) to the speed of light. Express your answer as a multiple of 𝑐. (d) Assume that the neutron star is uniform and calculate its density. Express your answer in kilograms per meter cubed.