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Which one of the following equations of motion represents simple harmonic motion? (where k, $k_0$, $k_1$ and a are all positive)
A
Acceleration = kx
B
Acceleration = $-k_0x + k_1x^2$
C
Acceleration = –k(x + a)
D
Acceleration = k(x + a)
Detailed Solution
A motion is simple harmonic if the acceleration is directly proportional to the displacement from a fixed (mean) position and is directed towards it: acceleration = $-\omega^2\times$(displacement from mean position).
Acceleration = –k(x + a): put $X = x + a$. Then acceleration = –kX, which is proportional to X and opposite to it. This is SHM about the mean position x = –a.
Acceleration = kx and acceleration = k(x + a): the acceleration is in the same direction as the displacement, so the particle moves farther away; there is no restoring effect.
Acceleration = $-k_0x + k_1x^2$: the acceleration is not linear in x, so the motion is not simple harmonic.
Hence the equation representing SHM is acceleration = –k(x + a).
Acceleration = –k(x + a): put $X = x + a$. Then acceleration = –kX, which is proportional to X and opposite to it. This is SHM about the mean position x = –a.
Acceleration = kx and acceleration = k(x + a): the acceleration is in the same direction as the displacement, so the particle moves farther away; there is no restoring effect.
Acceleration = $-k_0x + k_1x^2$: the acceleration is not linear in x, so the motion is not simple harmonic.
Hence the equation representing SHM is acceleration = –k(x + a).