Question

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HW 6.1. Briefly answer the following questions. a) A clock is mounted on the wall. As you look at it, what is the direction of the angular velocity vector of the second hand? What is the value of the angular acceleration of the second hand? b) If the angular acceleration of a rigid body is zero, what is the functional form of the angular velocity? c) What if another planet the same size as Earth were put into orbit around the Sun along with Earth. Would the moment of inertia of the system increase, decrease, or stay the same? d) Does increasing the number of blades on a propeller increase or decrease its moment of inertia, and why? e) Can you think of a body that has the same moment of inertia for all possible axes? If so, give an example, and if not, explain why this is not possible. Can you think of a body that has the same moment of inertia for all axes passing through a certain point? If so, give an example and indicate where the point is located. f) To maximize the moment of inertia of a flywheel while minimizing its weight, what shape and distribution of mass should it have? Explain. g) Is it possible to change the translational) kinetic energy of an object without changing its rotational kinetic energy? What about the reverse of it?

Answer 1

In a clock, there will a second's hand, a minute's hand, and an hour's hand. Seconds hand completes one revolution in 1 minute. This time taken would be same all the time. Similarly, the minutes hand completes one revolution in 1 hour. This time will also be these same for the minutes hand all the time. Angular displacement is the angle covered over the given time interval. The angular acceleration exists only when the speed of object revolving in a circle change with time. The angular displacement, angular velocity, and angular acceleration all are axial vectors and so they act along the axis of the rotation and their direction can be identified with the help of right hand rule or cross product rule. In the given case, the direction of the angular velocity must be acting in to the page based on the direction of revolution. The seconds hand usually revolves in clockwise direction and so use the right-hand rule that curl the fingers along the direction of rotation and the thumb will shows the direction of angular velocity. Thus, the angular velocity must be pointed in to the wall when looking from above. The angular acceleration of the seconds hand will be zero. This is because the speed of seconds hand is constant and so it does not change with time. Therefore, the tangential acceleration is zero, which in turn results in the zero angular acceleration. Thus, the angular acceleration of seconds hand is zero. (6) The angular acceleration could be zero only when the angular velocity is constant. This is because the angular acceleration can be defined as the time rate of change in the angular velocity. As long as the angular velocity is constant, the angular acceleration must be zero. So, the angular velocity need not be a function of time for the angular acceleration to be zero (uniform angular acceleration). Yes, the moment of inertia of the system of two planets around the sun must be increased. Explanation: The total moment of inertia must be as follows: (consider the earth and sun as solid spheres) Before another planet is placed in the earth's orbit: I segon = 1.u.ce + lewo.cae + Learn = {M R.? + M cover more? + M curth (Round even)? After another planet is placed in the earth's orbit: Igen = Lan cx +21eancu +21 cm =M_R..? +2{{M.czniPres?) +2(Mara ( Prava uza)") Thus, the moment of inertia increases.

(d) Increases Explanation: Consider each blade as solid rods. The moment of inertia of a solid rod rotating about an axis passing perpendicular to its length and from one its ends is, Iyed =ML If number of blades on a propeller is increased, then the net moment of inertia would also increase. If n number of blades introduced in a propeller, then the moment of inertia would be: The moment of inertia will not be same for all possible axes of an object. For example: Consider a solid sphere (symmetric). The moment of inertia of this object about an axis passing through its center of mass (or center) is mr?. If the axis is considered now tangential to its surface, then the moment of inertia about this axis would be - mr? + mr2 = -mr. For any object it is not possible that its moment of inertial is same for all possible axes. 5 The moment of inertia will be same for all possible axes of an object passing through the same point. For example: Consider a solid sphere (symmetric). The moment of inertia of this object about an axis passing through its center of mass (or center) is - mr. This axis can be considered in any possible direction but make sure that it should pass through its center of mass. From the axis of symmetry, the moment of inertia of this object would be same.