Simple Harmonic Motion. Why is Acceleration proportional to NEGATIVE displacement?

In summary: When the displacement is positive, the direction of the force is negative, and vice versa. Therefore, the acceleration is proportional to the negative displacement to account for this opposing force and maintain simple harmonic motion. This is represented by the negative sign in the formula for acceleration: a = -w^2x. In summary, in simple harmonic motion, the acceleration is proportional to the negative displacement because acceleration and displacement are vectors with opposite directions, and this is necessary to maintain the oscillating motion of the system.
  • #1
Yehia11
14
0
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!
 
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  • #2


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.
 
  • #3


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.
 
  • #4


Yehia11 said:
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.
 
  • #5


Yehia11 said:
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
 

1. What is Simple Harmonic Motion (SHM)?

Simple Harmonic Motion is a type of periodic motion in which an object oscillates back and forth around an equilibrium point. This motion is characterized by a restoring force that is directly proportional to the displacement of the object from its equilibrium position.

2. What are the factors that affect the period of SHM?

The period of SHM is affected by three main factors: the mass of the object, the stiffness of the spring or restoring force, and the amplitude of the oscillations. These factors determine how quickly the object will oscillate back and forth.

3. Why is acceleration proportional to the negative displacement in SHM?

In SHM, the acceleration of the object is always directed towards the equilibrium point. As the object moves away from the equilibrium point, the restoring force becomes stronger, causing the acceleration to increase in magnitude. When the object is moving towards the equilibrium point, the restoring force decreases, causing the acceleration to decrease. This relationship between acceleration and displacement is described by the equation a = -ω²x, where ω is the angular frequency and x is the displacement.

4. How does SHM relate to real-life examples?

SHM can be observed in many real-life examples, such as the motion of a pendulum, the vibration of a guitar string, or the motion of a mass attached to a spring. It can also be seen in the behavior of many mechanical and electrical systems, such as car suspensions and electronic circuits.

5. What is the significance of SHM in science and engineering?

SHM is a fundamental concept in science and engineering, as it helps us understand and analyze the motion of many physical systems. It is used to model and predict the behavior of objects in a wide range of applications, from simple pendulums to complex structures. SHM is also the basis for many important technologies, such as clocks, seismometers, and sensors.

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