Physics · Oscillations · NEET
Velocity is maximum at the mean position (x = 0), the centre of the motion. Here v_max = A*w. Velocity becomes zero at the extreme positions (x = +A and x = -A), because the body stops for a moment before turning back. Use v = w*root(A^2 - x^2): put x = 0 to get the biggest value, put x = A to get zero.
Acceleration is maximum at the extreme positions (x = +A or x = -A), where a_max = A*w^2. It is zero at the mean position (x = 0). This is the opposite of velocity. Use a = -w^2*x: bigger x means bigger acceleration, and x = 0 gives zero acceleration.
Velocity depends on how far the body is from the centre through v = w*root(A^2 - x^2), so it is largest at the centre. Acceleration depends directly on displacement through a = -w^2*x, so it is largest at the ends. That is why the body moves fastest in the middle but feels the strongest pull-back force at the ends.
Both start with amplitude A. Velocity has one power of w (v_max = A*w) and acceleration has two powers of w (a_max = A*w^2). Each time you go from displacement to velocity to acceleration you multiply by one more w. So displacement max = A, velocity max = A*w, acceleration max = A*w^2.
Divide a_max by v_max: (A*w^2) / (A*w) = w. So a_max / v_max = w (angular frequency). This is a quick shortcut in NEET numericals: if a question gives you both maximum values, w is simply their ratio.
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
Two identical point masses P and Q, suspended from two separate massless springs of spring constants k1 and k2, oscillate vertically. If their maximum speeds are the same, the ratio A_Q/A_P of the amplitude of mass Q to that of mass P is
Try the real previous-year questions from this chapter — each with the answer and a full solution.
v_max = A*w, where A is amplitude and w is angular frequency. It occurs at the mean position (x = 0).
a_max = A*w^2, where A is amplitude and w is angular frequency. It occurs at the extreme positions (x = +A or x = -A).
At the mean position velocity is maximum (A*w) and acceleration is zero. The body moves fastest here but feels no restoring force.
At the extreme position velocity is zero and acceleration is maximum (A*w^2). The body stops for an instant and feels the strongest pull back toward the centre.
Divide them: w = a_max / v_max. Since a_max = A*w^2 and v_max = A*w, the amplitude cancels and you are left with w.
