1/ A block of mass m = 1.8 kg is dropped from height h = 41 cm onto a spring of

Question

1/ A block of mass m = 1.8 kg is dropped from height h = 41 cm onto a spring of spring constant k = 2470 N/m (see the figure). Find the maximum distance the spring is compressed.

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2/ Tarzan, who weighs 669 N, swings from a cliff at the end of a convenient vine that is 15 m long (see the figure). From the top of the cliff to the bottom of the swing, he descends by 2.2 m. The vine will break if the force on it exceeds 1220 N. What would the greatest force on the vine be during the swing?

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3/ The boy in the figure here is initially seated on the top of a hemispherical ice mound of radius R = 13.9 m. He begins to slide down the ice, with a negligible initial speed. Approximate the ice as being frictionless. At what height does the boy lose contact with the ice?

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4/A single conservative force F(x) acts on a 4.3 kg particle that moves along an x axis. The potential energy U(x) associated with F(x) is given by
U(x) = -2.4xe-x/3
where x is in meters. At x = 1.1 m the particle has a kinetic energy of 2.7 J. (a) What is the mechanical energy of the system? (b) What is the maximum kinetic energy of the particle and (c) the value of x at which it occurs?

Explanation / Answer

1)

Using the law of conservation of energy,

Energy from block = Energy absorbed by spring

mg(0.41 + x) = (1/2)(k)x^2

where

m = mass of the block = 1.8 kg.
g = acceleration due to gravity = 9.8 m/sec^2 (constant)
x = compressed length of the spring
k = spring constant = 2470 N/m (given)

Substituting values,

1.8(9.8)(0.41 + x) = (1/2)(2470)(x^2)

16.76(0.41 + x) = 1235x^2

Rewriting

1235x^2 - 16.76x - 8.33 = 0

Using the quadratic formula,

x = 0.23 m = 23 cm.

2)

Conservation of energy means that

GPE lost = KE gained
m . g . ?h = 0.5 . m . v^2

so at the bottom of the swing his velocity is given by

v^2 = 2 . g . 2.2

(Leave it like that, no need to work it out )

At the bottom of the swing, the vine must exert two forces. One is just mg, the force needed to support Tarzanagainst gravity. The second is the centripetal force needed to change his velocity.( he was going down, now he's going to be going up). Since he is travelling in the arc of a circle, the size of this centripetal force is given by
F = m . v^2

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