Velocity In Shm

Question

A loudspeaker cone vibrates in simple harmonic motion with amplitude 4 cm and angular frequency 50 rad per second. What is its speed at the moment its displacement from the central position is 2 cm?

RankGuru · Accepted Answer

The speed of an SHM oscillator at displacement $x$ is given by $v = \omega\sqrt{A^2 - x^2}$, a relation that follows directly from energy conservation between kinetic and potential stores. This expression makes clear that the speed is largest at the centre and shrinks to zero exactly at the turning points where all the energy is potential. Converting units, $A = 0.04	ext{ m}$, $x = 0.02	ext{ m}$, and $\omega = 50	ext{ rad/s}$. Then $v = 50\sqrt{(0.04)^2 - (0.02)^2} = 50\sqrt{0.0016 - 0.0004} = 50\sqrt{0.0012} = 50 	imes 0.03464 \approx 1.73	ext{ m/s}$. The value $2.00	ext{ m/s}$ is the maximum speed $\omega A$, which would only occur at the mean position where $x = 0$. The value $1.00	ext{ m/s}$ wrongly uses $\omega x$ instead of the correct square-root expression. The value $0.87	ext{ m/s}$ halves the correct answer, a slip from misreading the amplitude. This is the standard NCERT velocity-displacement relation for SHM. As a plausibility check, the speed at $x = A/2$ must be less than the maximum $2.00	ext{ m/s}$ but still a large fraction of it, and $1.73	ext{ m/s}$, which equals $\frac{\sqrt{3}}{2}$ of the maximum, fits this expectation perfectly.

Velocity In Shm

Select the correct option:

🔒 Solution Hidden from View

More oscillations and waves Practice Questions

Comparing uniform circular motion with simple harmonic motion, the projection of a particle moving s...

Sound propagating through the air inside a long tube compresses and rarefies the medium along the di...

A stretched guitar string carries a tension of 80 N and has a linear mass density of 0.005 kg per me...

According to the Laplace correction, the speed of sound in a gas depends on which combination of gas...

Two waves of the same frequency travelling in a medium arrive at a point exactly out of phase with e...

A string fixed at both ends and one metre long vibrates in its fundamental mode, so what is the wave...

An organ pipe closed at one end and open at the other resonates in its lowest mode, so which harmoni...

A piano tuner sounds a reference fork of 256 hertz alongside a slightly mistuned string and hears fo...

An ambulance siren emitting 400 hertz approaches a stationary listener at 30 metres per second while...

When an oscillating particle in simple harmonic motion is at half of its amplitude, what fraction of...

A block of mass 2 kg attached to a light spring of force constant 200 N per metre oscillates on a fr...

Two identical springs each of force constant k are connected in parallel and support a mass m, so ho...

About This Question

Solution