A disk of radius 30 cm is spinning about it axis at 60rmp it deacelerate at a constant rate of 2.0 rad/s^2 until it reaches 20 rpm
A find the initial and final angular speed in radians per sec
B fine the time it takes to deacelerate from the initial to the final angular speed
Find how many reveloution are performed in this itime
Find the final tangential and centipetal accelTions
To answer these questions, we'll need to use some basic concepts of rotational motion.
Given:
- Radius of the disk (r) = 30 cm = 0.3 m
- Initial angular speed (ωi) = 60 rpm
- Final angular speed (ωf) = 20 rpm
- Angular acceleration (α) = -2.0 rad/s^2 (negative sign indicates deceleration)
A) To convert the angular speed from rpm to radians per second, we can use the conversion factor:
1 rpm = 2π radians per minute
= 2π/60 radians per second
= π/30 radians per second
Therefore, the initial and final angular speeds in radians per second are:
- Initial angular speed (ωi) = 60 rpm * (π/30 radians per second) = 2π radians per second
- Final angular speed (ωf) = 20 rpm * (π/30 radians per second) = 2π/3 radians per second
B) To find the time it takes to decelerate from the initial to the final angular speed, we can use the formula:
ωf = ωi + α * t
Rearranging the formula to solve for time (t):
t = (ωf - ωi) / α
Substituting the given values:
t = (2π/3 - 2π) / -2.0
Simplifying the equation:
t = (-4π/3) / -2 = 2π/3 seconds
Therefore, it takes 2π/3 seconds to decelerate from the initial to the final angular speed.
C) To find the number of revolutions performed during this time, we can use the formula:
θ = ωi * t + (1/2) * α * t^2
Here, θ represents the angle rotated during the time interval t. However, we want to find the number of revolutions (N), not the angle in radians. Since one revolution is equal to 2π radians, we can use the conversion factor:
1 revolution = 2π radians
Therefore, the number of revolutions is given by:
N = θ / (2π)
Substituting the values:
N = (ωi * t + (1/2) * α * t^2) / (2π)
= (2π * 2π/3 + (1/2) * (-2.0) * (2π/3)^2) / (2π)
Simplifying the equation:
N ≈ 4.29 revolutions
Therefore, approximately 4.29 revolutions are performed during this time.
D) To find the final tangential and centripetal accelerations, we can use the formulas:
Tangential acceleration (at) = α * r
Centripetal acceleration (ac) = ω^2 * r
Substituting the given values and converting the units to m/s^2:
Tangential acceleration (at) = (-2.0 rad/s^2) * 0.3 m = -0.6 m/s^2 (deceleration)
Centripetal acceleration (ac) = (2π/3 rad/s)^2 * 0.3 m = 1.25 m/s^2
Therefore, the final tangential acceleration is -0.6 m/s^2, indicating deceleration, and the final centripetal acceleration is 1.25 m/s^2.