# Constant of Integration when calculating displacement

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1. Jun 8, 2016

### RubiksMelia

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
s = ∫ v dt

3. 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?

2. Jun 8, 2016

### cnh1995

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.

Last edited: Jun 8, 2016
3. Jun 8, 2016

### RubiksMelia

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

4. Jun 8, 2016

### Delta²

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.

5. Jun 8, 2016

### cnh1995

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.

6. Jun 8, 2016

### Delta²

. 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).

7. Jun 8, 2016

### RubiksMelia

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.