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Is the acceleration of a particle moving with simple harmonic motion inversely proportional to the displacement of the particle from the mean position?
Acceleration is directly proportional to displacement in simple harmonic motion.
There are perhaps two good explanations for this, one technical and one intuitive.
First let us define simple harmonic motion.
When a particle moves in a straight line so that the displacement of the particle with time is exactly given by a simple sine (or cosine) of time, then that it is simple harmonic motion.
For example: x=A sine (w t) .
Answer 1: (In two steps)
(a) If we know position as a function of time, we know velocity is the time rate of change of position.
v = w A cosine (w t)
(b) If we know velocity as a function of time, we know acceleration is the time rate of change of velocity.
a = -w2 A sine (w t)
* So, acceleration is proportional to displacement, and a(t)=-w2 x(t).
Answer 2: (In three steps)
(a) Simple harmonic motion occurs when a mass on an ideal spring oscillates.
(b) From Newton's laws, we know that acceleration is directly proportional to force.
a=F/m
(c) We know the force of an ideal spring is proportional to displacement (F=-kx).
* So, acceleration is proportional to displacement, and a(t)= -k/m x(t).
(This also tells is that w2 =k/m.)
As a result, "acceleration is directly proportional to displacement in simple harmonic motion."
What else can I help you with?
How can you tell that a motion is simple harmonic?
A motion is simple harmonic if the acceleration of the particle is proportional to the displacement of the particle from the mean position and the acceleration is always directed towards that mean position.
When is acceleration greatest in SHM?
Acceleration is greatest at the extremities of the motion in simple harmonic motion (SHM). This occurs when the displacement is maximum and the restoring force is also maximum, resulting in the highest acceleration.
In simple harmonic motion the restoring force must be proportional to the?
displacement
Why is displacement directly proportional to acceleration?
Acceleration is given by the equation v-u/t, whereby v is the final velocity of a body, u is the initial velocity and t is the time. So for there to be acceleration, there has to be a change in initial velocity (the starting speed of the body) and the final velocity of the body or if there is no change it would be 0/t and hence the acceleration will be 0. And we know that with a change in speed there has to be a change in displacement i.e. s=d/t. So to sum all this up with an increase in acceleration there is an increase in distance. But this is a very tricky question because you asked about displacement and since displacement is a vector quantity, it has both magnitude and direction. So if a body accelerates forward, then its acceleration is directly proportional to its displacement. However, if the body stops and starts accelerating backwards, it becomes a whole new different story. Given that displacement has direction, if the body moves backwards, then the displacement will be negative, but the acceleration positive. So displacement is directly proportional to acceleration ONLY when a body is moving forwards, but displacement is indirectly proportional to acceleration when a body is moving backwards from it's point of rest. But for distance, given that it is a scalar quantity, it only has magnitude so it is not affected by the direction of movement of the body. So distance is directly proportional to acceleration when a body is moving both forwards and backwards.
Define simple harmonic motion . prove that vibrating motion of mass attached to spring is Simple harmonic motion?
Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. In the case of a mass attached to a spring, the motion is simple harmonic because the restoring force (provided by the spring) is directly proportional to the displacement from equilibrium (Hooke's Law) and acts in the opposite direction to the displacement, resulting in a sinusoidal motion.
How can you tell that a motion is simple harmonic?
A motion is simple harmonic if the acceleration of the particle is proportional to the displacement of the particle from the mean position and the acceleration is always directed towards that mean position.
How you called an object in harmonic motion?
When the acceleration is directly proportional to the displacement from a fixed point and always directed towards that fixed point then such an oscillation or vibration is said to be simple harmonic
When is acceleration greatest in SHM?
Acceleration is greatest at the extremities of the motion in simple harmonic motion (SHM). This occurs when the displacement is maximum and the restoring force is also maximum, resulting in the highest acceleration.
Is motion of swing an example of simple harmonic motion?
A body undergoes simple harmonic motion if the acceleration of the particle is proportional to the displacement of the particle from the mean position and the acceleration is always directed towards that mean. Provided the amplitude is small, a swing is an example of simple harmonic motion.
Can every oscillatory motion be treated as simple harmonic motion in the limit of small amplitude?
No. Simple harmonic motion requires that the acceleration is proportional to the displacement (and in the opposite direction). It is possible to have periodic motion where that is not the case.
In simple harmonic motion the restoring force must be proportional to the?
displacement
IS simple harmonic motion when there is a restoring force opposite and proportional to a displacement?
Yes.
Why is displacement directly proportional to acceleration?
Acceleration is given by the equation v-u/t, whereby v is the final velocity of a body, u is the initial velocity and t is the time. So for there to be acceleration, there has to be a change in initial velocity (the starting speed of the body) and the final velocity of the body or if there is no change it would be 0/t and hence the acceleration will be 0. And we know that with a change in speed there has to be a change in displacement i.e. s=d/t. So to sum all this up with an increase in acceleration there is an increase in distance. But this is a very tricky question because you asked about displacement and since displacement is a vector quantity, it has both magnitude and direction. So if a body accelerates forward, then its acceleration is directly proportional to its displacement. However, if the body stops and starts accelerating backwards, it becomes a whole new different story. Given that displacement has direction, if the body moves backwards, then the displacement will be negative, but the acceleration positive. So displacement is directly proportional to acceleration ONLY when a body is moving forwards, but displacement is indirectly proportional to acceleration when a body is moving backwards from it's point of rest. But for distance, given that it is a scalar quantity, it only has magnitude so it is not affected by the direction of movement of the body. So distance is directly proportional to acceleration when a body is moving both forwards and backwards.
Define simple harmonic motion . prove that vibrating motion of mass attached to spring is Simple harmonic motion?
Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. In the case of a mass attached to a spring, the motion is simple harmonic because the restoring force (provided by the spring) is directly proportional to the displacement from equilibrium (Hooke's Law) and acts in the opposite direction to the displacement, resulting in a sinusoidal motion.
How simple pendulum obeys simple harmonic motion?
The acceleration of the bob is directly proportional to the displacement and towards the vertical position.If x represents the angular displacement towards the right, from the vertical. and if x', x'' represent the derivatives, then x'' = -kx where k > 0. This is the characteristic differential equation for SHM.
Can a motion be oscillatory but not simple harmonic?
Yes, a motion can be oscillatory without being simple harmonic. Simple harmonic motion specifically refers to a type of oscillatory motion where the restoring force is directly proportional to the displacement. Other types of oscillatory motion can have different relationships between the restoring force and displacement, making them non-simple harmonic.
What is the mathematical relationship between force and displacement?
For a simple harmonic oscillator, the force is proportional to the displacement F=-kx, where F is the force, x is the displacement, and k is a positive proportionality constant commonly referred to as the spring constant
