Why is Acceleration Proportional to Negative Displacement in SHM?

[Yehia11]

In Simple harmonic motion: Acceleration = -W^2 x (W being omega, rads per sec. and x beind displacement (distance if you will) ) Omega is a constant


but why is acceleration proportional to the NEGATIVE displacement, in other words, why MINUS??

Help appreciated. Thanks in advance!

 

[banerjeerupak]

acceleration and displacement are vectors. they have direction. the acceleration is opposite in direction to the displacement and tries to bring it back. hence the minus sign to signify that the two vectors are opposite in direction.

 

[dacruick]

yea baner is correct. This is why in simple harmonic motion we have sinusoidal functions representing the displacement velocity and acceleration.

we know that the time derivative of displacement is velocity, and the time derivative of velocity is acceleration.

so if we take the derivative of sine we get a cosine. then we take the derivative of a cosine and we get -sine. That -sine is representative of acceleration, and if you notice it is in the opposite direction of displacement.

 

[Doc Al]

but why is acceleration proportional to the NEGATIVE displacement, in other words, why MINUS??

To add to what banerjeerupak described, you might find it helpful to consider Hooke's force law (which leads to simple harmonic motion): F = -kx. The negative sign is important. When the displacement is positive, the restoring force (and thus the acceleration) is negative.

 

[vin300]

but why is acceleration proportional to the NEGATIVE displacement,
Help appreciated. Thanks in advance!

Because the spring is elastic, it resists a change in shape