In figure (12−E3) k = 100 N m−1, M = 1 kg and F = 10 N. (a) Find the compression of the spring in the equilibrium position. (b) A sharp blow by some external agent imparts a speed of 2 m s−1 to the block towards left. Find the sum of the potential energy of the spring and the kinetic energy of the block at this instant. (c) Find the time period of the resulting simple harmonic motion. (d) Find the amplitude. (e) Write the potential energy of the spring when the block is at the left extreme. (f) Write the potential energy of the spring when the block is at the right extreme.
The answer of (b), (e) and (f) are different. Explain why this does not violate the principle of conservation of energy.
Figure
Answers
(a) 0.1 m
(b) 2.5 J
(c) π / 5 sec
(d) 0.2 m
(e) 4.5 J
(f) 0.5 J
Explanation:
=> According to the question,
Spring constant, k = 100 N/m,
Mass of the block, M = 1 kg
Force, F = 10 N
(a) The compression of the spring in the equilibrium position, x:
F=k/x
∴ x=F/k
=10/100
=0.1 m
(b) The sum of the potential energy of the spring and the kinetic energy of the block, E:
E = potential energy + kinetic energy
E = 1/2kx² + 1/2Mv²
= 1/2 * 100 * 0.1² + 1/2 * 1 * 2²
= 0.5 + 2
= 2.5 J
(c) The time period of the resulting SHM, T :
T = 2π√M/k
= 2*π*√1/100
= 2*π*1/10
= π /5 sec
(d) The amplitude, A:
Work done = change in mechanical energy.
Fx =
(10)x = 1/2*k(A+x)² − 2.5
10x = 1/2×100(A+0.1)² − 2.5
10x = 50(A+0.1)² − 2.5
10x = 50A² + 0.5 - 2.5
∴ 50A² + 0.5 = 2.5 + 10x
50A² + 0.5 + 10A = 2.5 + 10A
50A² = 2
A² = 2/50
A² = 1/25
A = 1/5
A = 0.2 m
(e) The potential energy of the spring when the block is at the left extreme, P.E. :
P.E.=1/2*k*(A + x)²
=1/2*100*(0.1 + 0.2)²
= 50 (0.3)²
=50*(0.09)
=4.5 J
(f) The potential energy of the spring when the block is at the right extreme, P.E. :
Distance between the two extremes = 2A
P.E. = 1/2k(A+x)² − F(2A)
= 4.5 - 10 (2 *0.2)
= 4.5 - 10*0.4
= 4.5 - 4
= 0.5 J
Different answers of (b), (e) and (f) do not violate the law of conservation of energy because the work is done by the external force of 10 N.
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