O8L Pcg 1. A ball is attached to a horizontal cord of length L whose other end i

Question

O8L Pcg 1. A ball is attached to a horizontal cord of length L whose other end is fixed, as shown above. The ball is released from rest. Answer all questions below in terms of given variables and fundamental constants. (This means you should NOTplugi values like 10 N/kg for g; your answers should NOT include units. Just variables.) (a) What approach may be used to solve for the speed of the ball at the lowest point of its path? Choose one and justify your answer. Newton's second law, kinematics, and circular motion must be used O The work-energy principle must be used (b) Derive an expression for the ball's speed at the lowest point of its path (c) A peg is located a distance 0.8L directly below the point of attachment of the cord. Derive an expression for the speed of the ball when it reaches the top of its circular path about the peg. (d) Compare the answers to (b) and (c). Explain which should be bigger using physical reasoning (i.e. a thought experiment); then show that your answers reflect your physical reasoning.

Explanation / Answer

1(a) The work energy principle must be used.

(b)At the height equal to the length L, the PE of the system is:

PE = m g L

From conservation of energy, at the bottm, while its moving PE gets converted to KE

KE = 1/2 m v^2

m g L = 1/2 m v^2

v = sqrt (2 g L)

(c)For the motion of the ball around the peg, using enwrgy conservation:

0 = change in KE + change in PE

0 = 1/2 m (v1^2 - v^2) + m g (h1 - h1)

1/2 m v1^2 = -g m (2(L - h)-L)

1/2 v1^2 = g (2h - L) = g (2 x 0.8L - L) = 0.6 L

v1 = sqrt (1.2 g L)

(d)We have, v = sqrt (2 g L) and v1 = sqrt (1.2 g L)

clearly, the first one is bigger this is because, the length is getting reduced.

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