A massless spring of constant k = 78.0 N/m is fixed on the left side of a level

Question

A massless spring of constant k = 78.0 N/m is fixed on the left side of a level track. A block of mass m = 0.50 kg is pressed against the spring and compresses it a distance of d, as in Figure P7.57. The block (initially at rest) is then released and travels toward a circular loop-the-loop of radius R = 1.5 m. 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

Explanation / Answer

k=78 N/m v(top)=velocity at the top of the loop; v(bottom)= velocity at bottom of the loop

m=0.5 kg

r=1.5 m

=0.36

l=2.5 m

g=9.8 m/s2

for the block to have a zero reaction force from the track at the uppermost part of the loop,
the velocity at the top =(rg) = 3.83 m/s

energy at the top = energy at the bottom of the loop

1/2 mv(top)2 + mg2r = 1/2 mv(bottom)2 (mg2r = potential energy, where r= loop radius)

solving the above equation for v(bottom), we get

v(bottom)=8.571 m/s

work done against friction = mgl = 0.36 * 0.5 * 9.8 * 2.5 = 4.41 joules

total energy during release = 4.41 + 1/2 * 0.5 * (8.571)2 = 22.77 joules

this much energy = 1/2kd2

solving the above equation for d, we get

d = 0.76m (answer)

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