A thin, uniform, rectangular sign hangs vertically above the door of a shop. The

Question

A thin, uniform, rectangular sign hangs vertically above the door of a shop. The sign is hinged to a stationary horizontal rod along its top edge. The mass of the sign is 2.40 kg and its vertical dimension is 50.0 cm. The sign is swinging without friction, becoming a tempting target for children armed with snowballs. The maximum angular displacement of the sign is 25.0° on both sides of the vertical. At a moment when the sign is vertical and moving to the left, a snowball of mass 570 g, traveling horizontally with a velocity of 160 cm/s to the right, strikes perpendicularly the lower edge of the sign and sticks there.

(a) Calculate the angular speed of the sign immediately before the impact.

_____rad/s

(b) Calculate its angular speed immediately after the impact.

____rad/s

(c) The spattered sign will swing up through what maximum angle?

____°

Explanation / Answer

a)

conservation of energy

Ei = Ef

Ei = M* g * ( 1 - cos(theta)) * L / 2

Ef = 0.5 * I * w^2

I = M * L^2 / 3

M * g * L * ( 1 - cos(theta)) / 2 = 0.5 * I * w^2

w = sqrt( 3 * g * ( 1 - cos(theta) ) / L

w = sqrt( 3 * 9.8 * ( 1 - cos(25deg) ) / 0.5) = 2.347 rad/sec

b)

conservation of angular momentum

Li = Lf

Li = m * v * R - M * R^2 * w = m * v * L - M * L^2 * w / 3

Li = 0.57 * 1.6 * 0.5 - 2.4 * 0.5^2 * 2.347 / 3

Li = 0.0134

then

Lf = (m * L^2 + M * L^2 /3) * w1

Lf = 0.3425 w1

by using both equation

w1 = m * v* L - M*L^2 *w /3 / ( m *L^2 + M *L^2 /3)

w1 = 0.0134 / 0.3425 = 0.04 rad/s

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