Pendulum Problem

# Pendulum Problem - m 1 v 1 i v 1 f = m 2 v 2 f(3 m 1 v 1 i...

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Mass m 1 , starting at a height h , swings down and collides with mass m 2 as shown in the figure below. Find the final velocity for m 1 . First we need to find the initial velocity of mass 1, v 1i . We do this using conservation of energy. The gravitational potential energy of mass 1 will be converted into kinetic energy just before it collides with mass 2. Equating these two energies, we find m 1 gh = 1 2 m 1 v 1 i 2 v 1 i = 2 gh Next, we use conservation of momentum and conservation of energy to solve for the final velocity of mass 1.

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Conservation of momentum tells us that m 1 v 1 i = m 1 v 1 f + m 2 v 2 f (1) Conservation of energy tells that 1 2 m 1 v 1 i 2 = 1 2 m 1 v 1 f 2 + 1 2 m 2 v 2 f 2 (2) We can rewrite these equations as
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Unformatted text preview: m 1 v 1 i ! v 1 f ( ) = m 2 v 2 f (3) m 1 v 1 i 2 ! v 1 f 2 ( ) = m 2 v 2 f 2 (4) If we solve (3) for v 2f and plug that into (4), we get m 1 v 1 i 2 ! v 1 f 2 ( ) = m 2 m 1 v 1 i ! v 1 f ( ) m 2 " # \$ % & ’ 2 or simplifying m 2 v 1 i 2 ! v 1 f 2 ( ) = m 1 v 1 i ! v 1 f ( ) 2 The difference of the squares can be rewritten as m 2 v 1 i ! v 1 f ( ) v 1 i + v 1 f ( ) = m 1 v 1 i ! v 1 f ( ) v 1 i ! v 1 f ( ) Canceling ( v 1i – v 1f ) from each side leaves us with m 2 v 1 i + v 1 f ( ) = m 1 v 1 i ! v 1 f ( ) Solving for v 1f , we get v 1 f = v 1 i m 1 ! m 2 m 1 + m 2 " # \$ % & ’...
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Pendulum Problem - m 1 v 1 i v 1 f = m 2 v 2 f(3 m 1 v 1 i...

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