A group of students observes that a wooden block (m = 0.40 kg) on the end of a string with a radius of 0.7 meters makes 8 rotations in 20.7 seconds when twirled.
What is the block's tangential (linear) speed?
What is the block's angular speed?
You have this equation:
Velocity = 2pi x radius / T
So...
V = 2pi x 0.7 meters / (20.7 sec / 8 rotations)
That should give you the right answer. As a note, T, the time it takes to complete a period, has a unit of sec / rotation in this case, even if it may be more intuitive to calculate for rotations / sec (what I did before I realized that was wrong).
To find the block's tangential (linear) speed, we can use the formula:
Tangential speed = (2 * π * radius * number of rotations) / time
1. We are given:
- Radius (r) = 0.7 meters
- Number of rotations = 8
- Time (t) = 20.7 seconds
Using the formula, we can calculate the tangential speed:
Tangential speed = (2 * π * 0.7 * 8) / 20.7
2. Now let's calculate the block's angular speed. The angular speed is the rate of change of the angle in radians per second.
Angular speed = (2 * π * number of rotations) / time
Using the given values, we can calculate the angular speed:
Angular speed = (2 * π * 8) / 20.7
Now, let's calculate both the tangential (linear) speed and the angular speed.
To find the block's tangential (linear) speed, we can use the formula:
v = ω * r
where:
v is the tangential speed,
ω is the angular speed, and
r is the radius of the circular path.
To find the block's angular speed, we can use the formula:
ω = θ / t
where:
ω is the angular speed,
θ is the angle rotated (in radians), and
t is the time taken for the rotations.
Given that the wooden block rotates 8 times in 20.7 seconds, we can calculate the angle rotated:
θ = 2 * π * n
where:
θ is the angle in radians, and
n is the number of rotations.
Now, let's calculate the tangential (linear) speed and the angular speed:
1. Tangential (Linear) Speed:
v = ω * r
First, calculate the angular speed:
ω = θ / t
= (2 * π * n) / t
= (2 * π * 8) / 20.7 rad/s (substituting the given values)
Next, calculate the tangential (linear) speed:
v = ω * r
= (2 * π * 8) / 20.7 * 0.7 m/s (substituting the given radius)
2. Angular Speed:
ω = θ / t
= (2 * π * n) / t
= (2 * π * 8) / 20.7 rad/s (substituting the given values)
The tangential (linear) speed is calculated to be (2 * π * 8) / 20.7 * 0.7 m/s, and the angular speed is (2 * π * 8) / 20.7 rad/s.