Define simple harmonic motion. show that in s.h.m., the acceleration is directly proportional to the displacement at any instant but opposite in direction.
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Simple harmonic motion is a special type of periodic motion or oscillatory motion in which restoring force is directly proportional to displacement of particle but directions are opposite . Trajectory of SHM is given by
y = Asin(ωt + Ф)
Here y is displacement of particle ,
ω is angular frequency , Ф is phase constant And A is amplitude of motion .
y = Asin(ωt + Ф) -----------------------(1)
differentiate with respect to t
dy/dt = Aωcos(ωt + Ф)
again differentiate with respect to t
d²y/dt² = -ω²Asin(ωt + Ф)
from equation (1),
d²y/dt² = -ω²y
d²y/dt² = acceleration , a
Then, a = -ω²y [ negative sign shows that acceleration is opposite with displacement]
Hence , it is clear that in SHM , acceleration is directly proportional to displacement at any instant but in opposite sign
y = Asin(ωt + Ф)
Here y is displacement of particle ,
ω is angular frequency , Ф is phase constant And A is amplitude of motion .
y = Asin(ωt + Ф) -----------------------(1)
differentiate with respect to t
dy/dt = Aωcos(ωt + Ф)
again differentiate with respect to t
d²y/dt² = -ω²Asin(ωt + Ф)
from equation (1),
d²y/dt² = -ω²y
d²y/dt² = acceleration , a
Then, a = -ω²y [ negative sign shows that acceleration is opposite with displacement]
Hence , it is clear that in SHM , acceleration is directly proportional to displacement at any instant but in opposite sign
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