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QZ_04_S

# QZ_04_S - Ph1a Solution to Quiz 4 Alejandro Jenkins Fall...

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Ph1a: Solution to Quiz 4 Alejandro Jenkins, Fall 2004 Problem 1 (6 points) - Collision on a Spring A linear spring has a free length D . When a mass m is hung on one end, the spring has an equilibrium length D + ` . While it is hanging motionless with an attached mass m , a second mass m is dropped from a height ` onto the ﬁrst one. The masses collide inelastically and stick together. (a) (1 point) What is the new equilibrium length of the spring? Solution: The equilibrium length of the spring is that for which the spring’s restoring force exactly cancels the weight of the attached mass. If a linear spring with elastic constant k is stretched by an amount ` (measured from its free length D ), then the restoring force is k` . If a mass m was attached, then, at equilibrium, k` = mg . If the weight is doubled, then the spring’s restoring force at its new equilibrium length must be twice what it was before: 2 mg = k 2 ` . The spring will now be stretched by 2 ` with respect to its free length. The total length of the spring at its new equilibrium is D + 2 ` . (b) (1 point) What is the period of the resulting motion? Solution: The spring’s stiﬀness, characterized by the elastic constant k , has not changed. The mass attached is now 2 m . Therefore the period is T = 2 π s 2 m k . But k is not speciﬁed in the problem, so we must express it in terms of other quantities given. Since the spring stretched by an amount ` under the weight of m , we have k` = mg , or k = mg/` . Therefore, T = 2 π s 2 ` g . (c) (2 points)

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QZ_04_S - Ph1a Solution to Quiz 4 Alejandro Jenkins Fall...

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