q3s - Physics 2211 Spring 2008 Quiz #3 Solutions Unless...

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Physics 2211 Quiz #3 Solutions Spring 2008 Unless otherwise directed, all springs, cords, and pulleys are ideal, and drag should be neglected. I . (16 points) The Earth is a planet with mass M and radius R . What would the period of the Earth’s rotation be if objects at the equator had an apparent weight that was one-third of the gravitational force on them? Express your answer in terms of any or all of M , R , and physical or mathematical constants. Note: Do NOT use the physical constant “ g ”, as it includes the eﬀect of the Earth’s actual period. . . . . . . . . . . . . . . . . . . . . . . . This is a Newton’s Second Law problem with Uniform Circular Motion. Sketch a Free Body Diagram of the object at the equator. Choose the positive direction to be in the direction of the centripetal acceleration, to- ward the center of the Earth. The only forces acting on the object are the force of gravity and a normal force. The normal force is the object’s apparent weight. The force of gravity on the object will be calculated using the Law of Universal Gravitation. The acceleration is a centripetal acceleration. Note that the speed of the object, v , is the circumference of the Earth, 2 πR , over the period, T . Letting M be the mass of the Earth, and m be the mass of the object, X F r = F G - n = ma r F G - 1 3 F G = m v 2 R G Mm R 2 - 1 3 G Mm R 2 = m R ± 2 πR T 2 Solve for T . Note that the mass of the object cancels. 2 3 G Mm R 2 = 4 2 R T 2 T 2 = 3 2 4 π 2 R 3 GM T = r 6 π 2 R 3 GM Quiz #3 Solutions Page 1 of 5

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1. (6 points) An object of mass M is attached to the end of a string and swings in a vertical circle having a radius r . At an instant when the string is at an angle θ above the horizontal, the tension in the string is T . What is the radial component of the weight? ( On Earth. ) . . . . . . . . . . . . . . . . . . . . . . . Sketch a Free Body Diagram of the object. Resolve the
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q3s - Physics 2211 Spring 2008 Quiz #3 Solutions Unless...

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