# Constant of Integration when calculating displacement

• RubiksMelia
In summary, the speed of a pendulum bob moving in simple harmonic motion is given by v = 1.26sin(2πt) where v is in m/s and t is time in seconds. At t=0, the pendulum is at the center and has zero velocity.f

## Homework Statement

The speed of a pendulum bob moving in simple harmonic motion is given by v = 1.26sin(2πt) where v is in m/s and t is time in seconds.

s = ∫ v dt

## The Attempt at a Solution

v = 1.26sin(2πt)

Integrating v yields

s = -0.2cos(2πt) + c

and solving for c where s, t = 0 yields c = 0.2 (as cos 2πt = 1 when t = 0), meaning that s = 0.2 - 0.2cos(2πt)

The answer in the back of the textbook seems to ignore this fact, simply stating that the displacement is equal to -0.2cos(2πt).

My question is this: does the constant of integration always matter when integrating like this to find displacements and velocities? Can it be "ignored"? Is the back of my textbook lying to me about the answer? and solving for c where s, t = 0 yields c = 0.2 (as cos 2πt = 1 when t = 0), meaning that s = 0.2 - 0.2cos(2πt)
You can see at t=0, the pendulum has zero velocity, meaning it is at one of the extremities. So, at t=0, the displacement should be maximum.

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You can see at t=0, the pendulum has zero velocity, meaning it is at one of the extremities. So, at t=0, the displacement should be maximum.

But, at t = 0, shouldn't the pendulum be at rest in the center (rather than the left or right extremity)?

The constant of integration matters and it is calculated by the initial conditions.
You seem to take as initial condition s(0)=0, but this is not correct and the reason is as cnh1995's post explains.

But, at t = 0, shouldn't the pendulum be at rest in the center (rather than the left or right extremity)?
In SHM, the pendulum is at the center when its velocity is maximum. Its velocity goes on decreasing till the extremity, where it becomes zero. In SHM, t=0 does not mean the pendulum is at the center.

But, at t = 0, shouldn't the pendulum be at rest in the center (rather than the left or right extremity)?
. This means that s(0)=0, v(0)=0 but this means that the pendulum will stay at center and do nothing (the force is zero at the center).

In SHM, the pendulum is at the center when its velocity is maximum. Its velocity goes on decreasing till the extremity, where it becomes zero. In SHM, t=0 does not mean the pendulum is at the center.
Thanks! I should have had a look at simple harmonic motion. Silly me for tackling a question before knowing what everything meant. Thanks to all for the help.