a) What is the tangential acceleration of a bug on the rim of a 6in. diameter disk if the disk moves from rest to an angular speed of 79 rev/min in 5 seconds?
b) One second after the bug starts from rest, what is its tangential acceleration? What is its centripetal acceleration?
To find the tangential acceleration of the bug on the rim of the disk, we need to use the following equations:
a) The angular speed (ω) conversion from rev/min to rad/s is given by:
ω = (2π/60) * N
where N is the number of revolutions per minute.
b) The formula for tangential acceleration is:
at = r * α
where at is the tangential acceleration, r is the radius of the disk, and α is the angular acceleration.
c) The formula for angular acceleration is:
α = (ωf - ωi) / t
where α is the angular acceleration, ωf is the final angular speed, ωi is the initial angular speed, and t is the time taken.
d) The formula for centripetal acceleration is:
ac = r * ω^2
where ac is the centripetal acceleration, r is the radius of the disk, and ω is the angular speed.
Let's calculate each part step-by-step:
a) First, convert the given angular speed from rev/min to rad/s:
ω = (2π/60) * 79 rev/min
ω = (2π/60) * 79 * 2π rad/s
ω ≈ 8.29 rad/s
b) Using the given time of 5 seconds, calculate the angular acceleration:
α = (ωf - ωi) / t
α = (8.29 rad/s - 0 rad/s) / 5 s
α ≈ 1.66 rad/s^2
c) Use the given radius of the disk of 6 inches to calculate the tangential acceleration:
at = r * α
at = (6 in.) * (1.66 rad/s^2)
at ≈ 9.96 in./s^2
d) To find the tangential acceleration and centripetal acceleration one second after the bug starts from rest, we need to know the initial angular speed ωi.
Since the bug starts from rest, ωi = 0 rad/s.
Using the given radius of the disk of 6 inches and the final angular speed from part a, we can now calculate the centripetal acceleration:
ac = r * ω^2
ac = (6 in.) * (8.29 rad/s)^2
ac ≈ 328.2 in./s^2
Therefore, one second after the bug starts from rest:
- The tangential acceleration is approximately 9.96 in./s^2.
- The centripetal acceleration is approximately 328.2 in./s^2.