Consider the track shown in the figure. The section AB is one quadrant of a circ

Question

Consider the track shown in the figure. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 2.8m long with a coefficient of kinetic friction ?k = 0.26. The section CD under the spring is frictionless. A block of mass 1.0 kgis released from rest at A. After sliding on the track, it compresses the spring by 0.35m . (Figure 1)

A) Determine the velocity of the block at point B. Express your answer to two significant figures and include the appropriate units.

   vB =___________

B) Determine the thermal energy produced as the block slides from B to C. Express your answer to two significant figures and include the appropriate units.

   E = ____________

C) Determine the velocity of the block at point C. Express your answer to two significant figures and include the appropriate units.

vC = ____________

D) Determine the stiffness constant k for the spring. Express your answer to two significant figures and include the appropriate units.

     k = ____________

Explanation / Answer

(a)
The potential energy at the top of the track, 2.0 m above point B is
PE = mgh = 1 x 9.8 x 2 = 19.6 J
At B, the kinetic energy, KE = PE = 19.6 J = 0.5mv^2
so v = sqrt[2 * 19.6 / 1]
v = 6.3 m/s

(b)
The friction force, F = 1 kg * 9.8m/s^2 * 0.26 = 2.548N
Thermal energy, ET = F * x = 2.548 * 2.8
ET = 7.134 J = 7.1 J (in two significant figures

(c)
The work done on from B to C subtracts from the kinetic energy at B
19.6 - 7.134 = 12.466 J = 0.5 * 1 * v^2
v = sqrt[2 * 12.466]
v = 5 m/s (approx)

(d)
The kinetic energy at C compresses the spring and stores that energy in the spring = 0.5kx^2
k = 12.466 / [(0.5)(0.35^2)
k = 203.53 N/m = 2 x 102 N/m

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