A circular disc with diameter of 11 cm rotates 280 rpm to 1400 rpm in 20 seconds. Then its slow down, α=-4.5rad/s2 in 8 seconds to angular velocity, ω1 rad/s. Determine
A. Angular acceleration and angular velocity ω1 rad/s
B. Total revolution been made
C. Centripetal acceleration and force at t= 15 seconds where mass is 75 g
To answer these questions, we'll need to use the formulas and equations related to rotational motion.
A. Angular acceleration and angular velocity:
We are given the initial and final angular velocities, as well as the time taken for the change in angular velocity. We can use the formula for angular acceleration (α) to find the angular acceleration:
α = (ωf - ωi) / t
where:
α = angular acceleration
ωf = final angular velocity
ωi = initial angular velocity
t = time taken
Substituting the given values, we have:
α = (1400 rpm - 280 rpm) / 20 s
Now, we need to convert the angular velocity from rpm to rad/s.
1 rpm = 2π rad/s (since there are 2π radians in one revolution)
Converting the angular velocities:
ωf = 1400 rpm * 2π rad/1 min * 1 min/60 s = 146.67 rad/s
ωi = 280 rpm * 2π rad/1 min * 1 min/60 s = 29.33 rad/s
Substituting these values into the formula for angular acceleration:
α = (146.67 rad/s - 29.33 rad/s) / 20 s
Calculating the angular acceleration:
α = 117.34 rad/s / 20 s
α ≈ 5.87 rad/s^2
So, the angular acceleration is approximately 5.87 rad/s^2.
Now, to find the angular velocity (ω1) at the end of the slow down period, we can use the formula:
ω1 = ωi + α * t
where:
ω1 = final angular velocity
ωi = initial angular velocity
α = angular acceleration
t = time taken
Substituting the given values, we have:
ω1 = 29.33 rad/s + (-4.5 rad/s^2) * 8 s
Calculating ω1:
ω1 = 29.33 rad/s - 36 rad/s
ω1 = -6.67 rad/s
So, the angular velocity ω1 is approximately -6.67 rad/s.
B. Total revolution made:
To find the total revolutions made, we need to calculate the total angle rotated in radians.
The formula to calculate the angle (θ) in radians is:
θ = ωi * t + (1/2) * α * t^2
where:
θ = angle in radians
ωi = initial angular velocity
α = angular acceleration
t = time taken
Substituting the given values, we have:
θ = 29.33 rad/s * 20 s + (1/2) * 5.87 rad/s^2 * (20 s)^2
Calculating θ:
θ = 586.6 rad + 587.5 rad
θ = 1174.1 rad
Since there are 2π radians in one revolution, we can calculate the number of revolutions (N) made:
N = θ / 2π
N = 1174.1 rad / 2π rad
N ≈ 186.99 revolutions
So, the total number of revolutions made is approximately 186.99 revolutions.
C. Centripetal acceleration and force at t = 15 seconds:
To calculate the centripetal acceleration and force, we need to consider the centripetal force formula:
Centripetal acceleration (a) = ω^2 * r
Centripetal force (F) = m * a
where:
a = centripetal acceleration
ω = angular velocity
r = radius
F = centripetal force
m = mass
Given that the diameter of the circular disc is 11 cm, the radius (r) is half the diameter:
r = 11 cm / 2 = 5.5 cm = 0.055 m
Converting the mass from grams to kilograms:
m = 75 g * 0.001 kg/g = 0.075 kg
We are asked to find the centripetal acceleration and force at t = 15 seconds.
Let's calculate that now:
Centripetal acceleration:
a = ω^2 * r
Substituting the given value of ω1:
a = (-6.67 rad/s) ^ 2 * 0.055 m
Calculating a:
a ≈ 0.302 m/s²
Centripetal force:
F = m * a
Substituting the given value of m and the calculated value of a:
F = 0.075 kg * 0.302 m/s²
Calculating F:
F ≈ 0.023 N
So, at t = 15 seconds, the centripetal acceleration is approximately 0.302 m/s² and the centripetal force is approximately 0.023 N.