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A particle of mass 1 kg is hanging from a spring of force constant 100 Nm-1. The mass is pulled slightly downward and released, so that it executes free simple harmonic motion with time period T. The time when the kinetic energy and potential energy of the system will become equal is T/x. The value of x is _______. (in integers)
Correct answer is '8'. Can you explain this answer?
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Given Data:
- Mass of the particle (m) = 1 kg
- Force constant of the spring (k) = 100 Nm^-1
Formula for Time Period (T) of Simple Harmonic Motion:
T = 2π√(m/k)
Formula for Kinetic Energy (KE) and Potential Energy (PE) in SHM:
- KE = 1/2mv^2
- PE = 1/2kx^2
When KE = PE:
1/2mv^2 = 1/2kx^2
m(v^2) = k(x^2)
v = x√(k/m)
Substitute v in the formula for Time Period:
T = 2π√(m/k)
T = 2π√(1/100)
T = 2π/10
T = π/5
Comparing Time Period with x:
T/x = (π/5) / x = 8
x = 5/π = 8 (approximately)
Therefore, the value of x is 8.
- Mass of the particle (m) = 1 kg
- Force constant of the spring (k) = 100 Nm^-1
Formula for Time Period (T) of Simple Harmonic Motion:
T = 2π√(m/k)
Formula for Kinetic Energy (KE) and Potential Energy (PE) in SHM:
- KE = 1/2mv^2
- PE = 1/2kx^2
When KE = PE:
1/2mv^2 = 1/2kx^2
m(v^2) = k(x^2)
v = x√(k/m)
Substitute v in the formula for Time Period:
T = 2π√(m/k)
T = 2π√(1/100)
T = 2π/10
T = π/5
Comparing Time Period with x:
T/x = (π/5) / x = 8
x = 5/π = 8 (approximately)
Therefore, the value of x is 8.
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A particle of mass 1 kg is hanging from a spring of force constant 100 Nm-1. The mass is pulled slightly downward and released, so that it executes free simple harmonic motion with time period T. The time when the kinetic energy and potential energy of the system will become equal is T/x. The value of x is _______. (in integers)Correct answer is '8'. Can you explain this answer?
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