Suppose that a disk rotates through two revolutions in 5.5 seconds.
(a) What is its displacement in radians in this time?
(b) What is its average rotational velocity in rad/s?
(a) Well, if the disk completes two full revolutions, that means it travels 2 * 2π radians. So, the displacement in radians would be 4π radians.
(b) To find the average rotational velocity, we need to divide the displacement (4π radians) by the time (5.5 seconds). Let's do some quick math, 4π divided by 5.5 is approximately 2π/2.75, which simplifies to about 0.73 rad/s.
So, the average rotational velocity of the disk is about 0.73 rad/s. It's spinning faster than a confused squirrel trying to find its acorns!
To find the displacement in radians, we need to know the number of radians in one revolution.
One revolution is equivalent to 2π radians because the circumference of a circle is 2π times the radius, and one revolution covers the full circumference.
(a) Displacement in radians = Number of revolutions x Number of radians in one revolution
= 2 revolutions x 2π radians/revolution
= 4π radians
Therefore, the displacement in radians is 4π radians.
To find the average rotational velocity in rad/s, we use the formula:
Average rotational velocity = Total displacement in radians / Total time taken
= 4π radians / 5.5 seconds
(b) Average rotational velocity = (4π / 5.5) radians/second
Therefore, the average rotational velocity is (4π / 5.5) radians/second.
To find the answers, we need to use the formulas relating rotational motion, time, and angular displacement.
(a) The angular displacement (θ) is given by the formula:
θ = 2πn,
where n is the number of revolutions.
Given that the disk rotates through two revolutions, we can substitute n = 2 into the formula:
θ = 2π(2) = 4π radians
Therefore, the displacement of the disk in radians is 4π radians.
(b) The average rotational velocity (ω) is calculated by dividing the angular displacement by the time taken:
ω = θ / t
Substituting the values:
ω = 4π / 5.5
Calculating this further, we get:
ω ≈ 2.2832 rad/s
Therefore, the average rotational velocity of the disk is approximately 2.2832 rad/s.
a.)2π radians = 1 revolution
4π radians = 2 revolutions
4*π=12.56637061 rad
b.) (2rev/ 5.5sec)(2π radians/revolution)
(0.363636)(6.283185301)
= 2.284794657