AUTOMATIC 5 STARS FOR CLEAR AND CONCISE ANSWER WITH F.B.D (IF NECESSARY) ! Formu

Question

AUTOMATIC 5 STARS FOR CLEAR AND CONCISE ANSWER WITH F.B.D (IF NECESSARY) !

Formulae Delta U = -W U(y) = mgy U(x) = 1/2kx2 F(x) = dU(x)/dx A block of mass m is dropped onto the top of a coil spring of constant k, which has its cylindrical axis oriented vertically. Consequently, the spring slows and momentarily stops the mass, at which point, the spring has been compressed by a distance d. a) While the spring is being compressed, show that the ratio of the work done by the spring to the work done by gravity is given by Show that the block's speed just before it meets the spring is given by If the block is dropped from a greater height so that its speed is doubled just before it meets the spring, show that the spring will then be compressed by a distance

Explanation / Answer

W spring = -1/2 k d^2
W gravity = m g d

a) Ws/Wg = -1/2 kd^2/(m g d) = -kd/(2 mg)

b)
Ei = Ef
1/2 mv^2 + mg d = 1/2 kd^2

v^2 + 2 g d = kd^2/m

v = sqrt( kd^2/m - 2gd)

c)

m g D + 1/2 m (2v)^2 = 1/2 k D^2

m g D + 2 m( kd^2/m - 2 g d) = 1/2 k D^2

-1/2 k D^2 + m g D + ( 2 kd^2 - 4 m g d) = 0

solve for D with quadratic equation

D = 1/k (mg + sqrt(m^2g^2 -8mg kd + 4 k^2 d^2))

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