A particle of mass 0.20\text{ kg} executes simple harmonic motion along a straight line. W… — JAMB UTME

A particle of mass $0.20\text{ kg}$ executes simple harmonic motion along a straight line. When its displacement from the equilibrium position is $0.03\text{ m}$ , its speed is $0.16\text{ m/s}$ . When its displacement is $0.04\text{ m}$ , its speed is $0.12\text{ m/s}$ . What is the total mechanical energy of the particle in millijoules ( $\text{mJ}$ )?

Answer: 4 mJ

Answer

The total mechanical energy of the particle is

4\text{ mJ}

Using the relation

v^2 = \omega^2(A^2 - x^2)

for the two given state points

(0.03\text{ m}, 0.16\text{ m/s})

and

(0.04\text{ m}, 0.12\text{ m/s})

forms a set of simultaneous equations. Subtracting them yields

\omega^2 = 16\text{ rad}^2/\text{s}^2

, leading to

A^2 = 0.0025\text{ m}^2

. Substituting these values into

E = \frac{1}{2}m\omega^2 A^2

gives

E = 0.004\text{ J}

, which converts to

4\text{ mJ}

Step-by-Step Solution

Set up kinematic equations for both displacement points using

v^2 = \omega^2(A^2 - x^2)

0.0256 = \omega^2(A^2 - 0.0009)

and

0.0144 = \omega^2(A^2 - 0.0016)

The equation relates linear speed, angular frequency, amplitude, and instantaneous displacement in SHM.

Subtract the two simultaneous equations to eliminate

A^2

and find

\omega^2

0.0112 = 0.0007\omega^2 \implies \omega^2 = 16\text{ rad}^2/\text{s}^2

Eliminating amplitude isolates the angular frequency squared.

Determine

A^2

by substituting

\omega^2 = 16

back into one of the state equations.

A^2 = 0.0025\text{ m}^2 \implies A = 0.05\text{ m}

Amplitude is required to calculate the maximum potential or total mechanical energy.

Calculate total mechanical energy

E = \frac{1}{2}m\omega^2 A^2

and convert to millijoules.

E = \frac{1}{2} \times 0.20 \times 16 \times 0.0025 = 0.004\text{ J} = 4\text{ mJ}

Total energy in SHM is constant and proportional to mass, square of angular frequency, and square of amplitude.

Key Concept

Conservation of energy and phase-space relationship between velocity and displacement in simple harmonic motion.

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