# Maximum Velocity When Doubling Amplitude & Period of SHM

In summary, when doubling both the amplitude and period of a SHM, the maximum velocity will remain unchanged as the sine component of the equation does not affect it. The maximum velocity is determined by the absolute value of the sine, so doubling or halving it will not change the maximum velocity.
what happens to the maximum velocity if you double both Amplitude and period of a SHM?

the equation is

x(t) = A cos (wt + f)
v(t) = -A w sin (wt + f)

f is phase constant, w = angular velocity

i know if i double A and T, the first part of the v(t) equation will be like:
-2A * 2pi/2T = -A * w, not changed, but what about that part in the sine ?

You're asked about the maximum velocity.That should happen when the sine is either +1 or -1 (abosolute value,nevermind the sense)...Therefore,it's not a factor to this problem...

Daniel.

If you double both the amplitude (A) and the period (T) of a simple harmonic motion (SHM), the maximum velocity will remain unchanged. This can be seen by plugging in the new values into the equation for velocity (v = -Aωsin(ωt + φ)). The first part of the equation, -Aω, will remain the same since the amplitude is doubled and the angular velocity (ω) is halved. However, the second part of the equation, sin(ωt + φ), will also remain the same since the period is doubled, resulting in the same value for ωt + φ. Therefore, the maximum velocity will not be affected and will remain the same as before.

## 1. What is simple harmonic motion (SHM)?

Simple harmonic motion is a type of periodic motion in which an object moves back and forth along a straight line with a constant frequency and amplitude. This motion is produced by a restoring force that is directly proportional to the displacement of the object from its equilibrium position.

## 2. How does doubling the amplitude of SHM affect the maximum velocity?

Doubling the amplitude of SHM will also double the maximum velocity. This is because the velocity of an object in SHM is directly proportional to the amplitude of its motion. When the amplitude is doubled, the object will cover a greater distance in the same amount of time, resulting in a higher maximum velocity.

## 3. What happens to the period of SHM when the amplitude is doubled?

Doubling the amplitude of SHM will not affect the period of the motion. The period of SHM is determined by the mass of the object and the force acting on it, not the amplitude. Therefore, the period will remain the same regardless of any changes in amplitude.

## 4. Can the maximum velocity in SHM exceed the maximum velocity of a freely falling object?

No, the maximum velocity in SHM cannot exceed the maximum velocity of a freely falling object. In SHM, the maximum velocity is determined by the amplitude and the period of the motion. The maximum velocity of a freely falling object is determined by the acceleration due to gravity, which is a constant value. Therefore, the maximum velocity in SHM will always be lower than the maximum velocity of a freely falling object.

## 5. Is there a relationship between the amplitude and period of SHM?

Yes, there is an inverse relationship between the amplitude and period of SHM. This means that as the amplitude increases, the period decreases and vice versa. This relationship is governed by the equation T = 2π√(m/k), where T is the period, m is the mass of the object, and k is the spring constant. As the amplitude increases, the value of k (which is directly proportional to the amplitude) increases, resulting in a decrease in the period.

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