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Unformatted text preview: Physics 2211 Quiz #3 Solutions Fall 2007 g Earth = 9 . 8m / s 2 Unless otherwise directed, all springs, cords, and pulleys are ideal, and drag should be neglected. I . (16 points) A ball of mass m is placed at a height L in the frictionless hollow cone shown. It is put into uniform circular motion. What angular velocity, , is required to keep the ball at the height L ? Express your answer in terms of any or all of m , L , , and physical or mathematical constants. ( On Earth. ) . . . . . . . . . . . . . . . . . . . . . . . Use Newtons Second Law. The vertical, or z , direction, will yield an expression for the normal force. X F z = n z- w = ma z = 0 n sin / 2 = mg n = mg sin( / 2) In the radial, or r , direction, the acceleration is a centripetal acceleration. X F r = n r = ma r = m v 2 R n cos( / 2) = mR 2 Note that R/L = tan( / 2), so R = L tan( / 2). Substituting this expression for R and the expression for n above n cos( / 2) = mR 2 mg sin( / 2) cos( / 2) = m [ L tan( / 2)] 2 mg tan( / 2) = m [ L tan( / 2)] 2 Solving for , = p g/L tan( / 2) Quiz #3 Solutions Page 1 of 6 II . (16 points) A ferry boat travels from point A to a dock B across the river in 40s. The width of the river is 200m and the dock B is down stream. The river flows at 2m/s. The speed of the ferry through the water is 10m/s. What is the velocity of the ferry ( i.e. , magnitude and direction, ) relative to the shore? . . . . . . . . . . . . . . . . . . . . . . . The velocity of the ferry through the water, ~v fw , which has a magnitude of 10m/s, can be resolved into two components, one across the river, v fw A , and one down the river, v fw D . The value of the component across the river is v fw A = D/ t = 200m / 40s = 5m / s The value of the component down the river can be found with the Pythagorean Theorem. v 2 fw = v 2 fw A + v 2 fw D v fw D = q v 2 fw- v 2 fw A v fw D = q (10m / s) 2- (5m / s) 2 = 8 . 7m / s The velocity of the ferry with respect to the shore, ~v fs is the sum of the velocity of the ferry with respect to the water, ~v fw...
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