To determine the velocity of a rotating platter given its period and radius, follow these steps:
1. Understand the problem's parameters:
- The period [tex]\(T\)[/tex] is 2.93 seconds. This is the time it takes for the platter to make one complete rotation.
- The radius [tex]\(r\)[/tex] of the platter is 12.8 centimeters.
2. Calculate the circumference of the platter:
The circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[
C = 2 \pi r
\][/tex]
where [tex]\(\pi\)[/tex] (pi) is approximately equal to 3.14159.
3. Substitute the known radius into the circumference formula:
[tex]\[
C = 2 \pi \times 12.8 \, \text{cm}
\][/tex]
Through the calculation, we find:
[tex]\[
C \approx 80.4247719318987 \, \text{cm}
\][/tex]
4. Determine the velocity:
The velocity [tex]\(v\)[/tex] of the rotating platter can be found by dividing the circumference by the period. This is because velocity is the distance traveled per unit of time.
[tex]\[
v = \frac{C}{T}
\][/tex]
where [tex]\(C\)[/tex] is the circumference and [tex]\(T\)[/tex] is the period.
5. Substitute the known values for circumference and period into the velocity formula:
[tex]\[
v = \frac{80.4247719318987 \, \text{cm}}{2.93 \, \text{s}}
\][/tex]
By doing the division, we find:
[tex]\[
v \approx 27.44872762180843 \, \text{cm/s}
\][/tex]
Therefore, the velocity of the rotating platter is approximately:
[tex]\[
v \approx 27.44872762180843 \, \text{cm/s}
\][/tex]
