Physics · Oscillations · NEET
Start from x = A sin(wt + phi). Velocity is the time-derivative: v = wA cos(wt + phi). Now use the identity cos^2 + sin^2 = 1, so cos(wt + phi) = root(1 - sin^2(wt + phi)) = root(1 - (x/A)^2). Put this in: v = wA root(1 - x^2/A^2) = w root(A2 - x2). No calculus is needed to use it - just remember the final form.
Put x = 0 in v = w root(A2 - x2). You get v = w root(A2) = wA, the largest value. Put x = A and you get v = w root(A2 - A2) = 0. So the particle moves fastest as it crosses the center and stops for an instant at the turning points before coming back.
v = w root(A2 - x2) always comes out positive, so it gives magnitude (speed) only. The real velocity can be plus or minus because the particle passes each point x twice - once going right, once going left. Use the time form v = wA cos(wt + phi) if you need the sign; use the displacement form when the question only asks for speed at a given x.
Use v = wA cos(wt + phi) when the question gives you time t. Use v = w root(A2 - x2) when the question gives you a position x and asks for the speed there - it saves you from first finding t. Both describe the same motion.
Square both sides: v^2 = w^2(A2 - x2). Multiply by (1/2)m and use w^2 = k/m. You get kinetic energy KE = (1/2)m w^2 (A2 - x2) = (1/2)k(A2 - x2). That is exactly total energy (1/2)kA2 minus potential energy (1/2)kx2 - so the velocity-displacement relation is just energy conservation in disguise.
A particle executes linear simple harmonic motion with an amplitude of 3 cm. When the particle is at 2 cm from the mean position, the magnitude of its velocity equals the magnitude of its acceleration. Its time period (in seconds) is
Try the real previous-year questions from this chapter — each with the answer and a full solution.
It is v = w root(A2 - x2), giving the speed of a particle at displacement x from the mean position, where w is angular frequency and A is amplitude.
Maximum velocity is v = wA, reached at the mean position where x = 0.
At the extreme position x = A, the speed is zero because root(A2 - A2) = 0. The particle momentarily stops there.
No. It gives only the magnitude (speed). Use v = wA cos(wt + phi) if you need whether the particle moves in the plus or minus direction.
Squaring gives v^2/(wA)^2 + x^2/A^2 = 1, an ellipse. So the v-x graph is an ellipse with x-intercepts at plus or minus A and v-intercepts at plus or minus wA.
