# circular motion and energy

**1. The problem statement, all variables and given/known data**

A massless spring of constant k is fixed on the left side of a level track. A block of mass m is pressed against the spring and compresses it a distance d, as shown in the figure. The block (initially at rest) is then released and travels toward a circular loop-the-loop of radius R. The entire track and the loop-the-loop are frictionless, except for the section of track between points A and B. Given that the coefficient of kinetic friction between the block and the track along AB is k, and that the length of AB is L. The gravitational acceleration is g.

a) Express the minimum speed of the mass at point C, vC, in terms of symbols given such that it can make through the loop-the-loop (i.e. remain in contact with the track).

(b) Hence or otherwise, determine the minimum compression, d, of the spring for such situation.

**2. Relevant equations**

Fnet=ma,

ac=v^2/R,

Total energy = KE + PE + friction

Spring energy = -kx

**3. The attempt at a solution**

a) Fnet=ma

N+mg=m(ac)

2mg=m(v^2/R)

2g=(v^2/R)

vc=sqt(2gR)

b) -kd=1/2m(vc)^2 + mgR + k

d= (1/2m(vc)^2 + mgR + k )/-k

Are these the correct approaches, please?

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