a mass being held by a string moves with speed v on a circular path of radius r, and has a period of revolution T. if it was desired to travel one revolution in twice as much time as before, what would be the radius of the path? what if the time was halved?
To find the radius of the circular path in each case, we can use the formulas related to the period and speed of a circular motion.
Let's first tackle the question of traveling one revolution in twice as much time as before.
1. Traveling one revolution in twice as much time:
If the original period of revolution is T, then according to the given condition, the new period will be 2T.
The period of revolution is the time taken to complete one full revolution. It is related to the speed by the formula:
T = (2πr) / v,
where r is the radius of the circular path and v is the speed of the mass.
Since we want to find the new radius, we need to rearrange the formula to solve for r:
r = (T * v) / (2π).
Substituting the new period 2T, we have:
r = (2T * v) / (2π) = (T * v) / π.
Therefore, if the time is doubled, the radius of the circular path will be (T * v) / π.
Now let's consider halving the time to travel one revolution:
2. Traveling one revolution in half the time:
If the original period of revolution is T, then according to the given condition, the new period will be T/2.
Using the same formula T = (2πr) / v, rearranging it to solve for r, we have:
r = (T * v) / (2π).
Now, substituting the new period T/2 into the equation:
r = ((T/2) * v) / (2π) = (T * v) / (4π).
Therefore, if the time is halved, the radius of the circular path will be (T * v) / (4π).
So, to summarize:
- If the time is doubled, the radius will be (T * v) / π.
- If the time is halved, the radius will be (T * v) / (4π).