Chapter 08, Problem 053 In the figure, a 3.3 kg block is accelerated from rest b

Question

Chapter 08, Problem 053 In the figure, a 3.3 kg block is accelerated from rest by a compressed spring of spring constant 630 N/m. The block leaves the spring at the spring's relaxed length and then travels over a horizontal floor with a coefficient of kinetic friction k = 0.243. The frictional force stops the block in distance D = 7.5 m, what are (a) the increase in the thermal energy of the block-floor system, (b) the maximum kinetic energy of the block, and (c) the original compression distance of the spring? No friction- (uk) (a) Number (b) Number (c) Number Units Units Units

Explanation / Answer

Mass of block M =3.3 kg
Spring constant k = 630 N/m
Coefficient of kinetic friction u = 0.243
Let S = distance travelled by the block after the spring comes to relaxed length.
S = 7.5 m

i)  

Normal force Fn = Mg = 3.3 * 9.8 = 32.373 N
Friction force Ff = u*Fn = 0.243 * 32.373 = 7.866 N
Friction is opposite to displacement.
Work done by friction = -Ff * S = -7.866 * 7.5 = -59 J
Increase in thermal energy = - work done by friction = -(-59) = 59 J

b) Block has maximum KE when the spring's potential energy = 0 i.e. when the spring is in relaxed length.
Let Kmax = maximum KE of the block
When the spring is in relaxed length, then
KE of the system = Kmax
PE of the system = 0

When the block stops, then
KE of the system = 0
PE of the system = 0
Therefore change in mechanical energy=(0+0)-(Kmax +0) = -Kmax
But change in mechanical energy = work done by friction
-Kmax = -59 J
Kmax = 59 J
Ans: 59 J

c) Let x = compression in the spring
By conservation of energy
1/2 kx2 = Kmax
1/2 * 630 * x2 = 59
x=0.4328m

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