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A net torque is analogous to a net force when examining its influence on motion. Newton's First Law (adapted to torque) In the absence of a net torque, an object will continue with its present rotational (angular) velocity. Newton's Second Law (adapted to torque) If there is a net torque, the object will have a rotational (angular) acceleration according to t=I*a. Newton's Third Law (remains unchanged for torque) When one object applies a force on another object the second object applies an equal and opposite force back on the first object. Equations of Motion for Constant Linear Acceleration and for Constant Angular Acceleration Suppose you are a visitor on another planet and observe the setting sun. You notice that your little finger, which is 1 cm wide, just covers the sun when you extend your arm out and hold your finger 1 m away from your eyes. The bottom edge of the sun begins to dip below the horizon, and 5 minutes later the sun completely disappears. Estimate the length of the day on this planet. The Crab nebula resulted from a supernova explosion seen in the year 1054. In addition to the gaseous debris seen here, the explosion left a spinning neutron star (pulsar) at its center. This pulsar has a period of rotation of T = 0.033 s that is increasing at the rate of 1.26x10 -5 s/y. Assuming a constant angular acceleration, in how many years from now will the pulsar stop rotating? What was the initial T for the pulsar? This massive shield door at a neutron test facility at Lawrence Livermore Laboratory is the world's heaviest hinged door. The door has a mass of 44,000 kg, a rotational inertia about a vertical axis through its huge hinges of 8.7x10 4 kg·m2, and a (front) face width of 2.4 m. Neglecting friction, what steady force, applied at its outer edge and perpendicular to the plane of the door, can move it from rest through an angle of 90° in 30 s? A block with mass m = 1.2 kg hangs from a massless cord that is wrapped around the rim of a uniform disk of mass M = 2.5 kg and radius R = 20 cm, mounted on a fixed horizontal axle. Assuming the cord does not slip and there is no friction at the axle, find the acceleration of the falling block and the tension in the cord. Fnet,y = may tnet = Ia at = ar A uniform ball, of mass M = 6.00 kg and radius R, rolls smoothly from rest down a ramp at angle q = 30.0°. What is its speed at the bottom, a vertical distance h = 1.20 m below? What are the magnitude and direction of the friction force on the ball as it rolls down the ramp? If the turntable’s moment of inertia is 5 times that of the record, what is the rotational speed of the record-turntable system after the collision? For a classroom demonstration, Cameron sits on a piano stool holding a sizable mass in each hand. Initially, the student holds his arms outstretched and spins about the axis of the stool with an angular speed of 3.74 rad/s. The moment of inertia in this case is 5.33 kg*m 2. While still spinning, the student pulls his arms in to his chest reducing the moment of inertia to 1.6 kg*m2. What is Cameron’s angular speed now? The spinning ice-skater Why does the cat always land on its feet? Frontside 180 More examples… The wheel shown has 8 equally spaced spokes and a radius of 30 cm. It is mounted on a fixed axle and is spinning at 2.5 rev/s. You want to shoot a 20-cm-long arrow parallel to this axle and through the wheel without hitting any of the spokes. Assume that the arrow and the spokes are very thin. (a) What minimum speed must the arrow have? (b) Does it matter where between the axle and rim of the wheel you aim? If so, what is the best location? What’s the acceleration of the falling hand? Will the ball land into the cup? In early 1985, Test Devices, Inc. (www.testdevices.com) was spin-testing a sample of a solid steel rotor (a disk) of mass M = 272 kg and radius R = 38.0 cm. When the sample reached an angular speed w of 14000 rev/min, the test engineers heard a dull thump from the test system, which was located one floor down and one room over from them. Just how much energy was released in the explosion of the rotor? Damage caused by the explosion included: •lead bricks thrown out in the hallway leading to the test room •a door to the room had been hurled into the adjacent parking lot •one lead brick shot from the test site through the wall of a neighbor's kitchen •the structural beams of the test building had been damaged •the concrete floor beneath the spin chamber had been shoved downward by about 0.5 cm •the 900-kg lid blown upward through the ceiling and then crashed back onto the test equipment An early method of measuring the speed of light makes use of a rotating slotted wheel. A beam of light passes through a slot at the outside edge of the wheel, travels to a distant mirror, and returns to the wheel just in time to pass through the next slot in the wheel. In one such setup, the wheel has a radius of 5.0 cm, has 500 slots at its edge, and the mirror is L = 500 m from the wheel. If the wheel turns at a rate of 3.8x103 rad/s and the light beam passes through two neighboring slots, what is the measured speed of light? In early 1985, Test Devices, Inc. (www.testdevices.com) was spin-testing a sample of a solid steel rotor (a disk) of mass M = 272 kg and radius R = 38.0 cm. When the sample reached an angular speed w of 14000 rev/min, the test engineers heard a dull thump from the test system, which was located one floor down and one room over from them. Evidently quite a bit structural damage had been caused by the explosion. How much energy was released in the explosion of the rotor? Damage caused by the explosion included: •lead bricks thrown out in the hallway leading to the test room •a door to the room had been hurled into the adjacent parking lot •one lead brick shot from the test site through the wall of a neighbor's kitchen •the structural beams of the test building had been damaged •the concrete floor beneath the spin chamber had been shoved downward by about 0.5 cm •the 900-kg lid blown upward through the ceiling and then crashed back onto the test equipment