A wheel starts from rest and rotates with constant angular acceleration and reaches an angular speed of 12.0 rad/s in 3.00 s. Find the angle (in radians) throuh which it rotates in this time.
clearly a = (12rad/s)/(3s) = 4 rad/s^2
As with linear motion, s = 1/2 at^2, so it traveled through
1/2 * 4 * 9 = 18 rad
To find the angle through which the wheel rotates, we can use the equation for angular displacement:
θ = ω₀t + (1/2)αt²
Where:
θ is the angular displacement
ω₀ is the initial angular speed (which is 0 because the wheel starts from rest)
t is the time
α is the angular acceleration
Given:
ω = 12.0 rad/s (final angular speed)
t = 3.00 s (time)
Since the wheel starts from rest, the initial angular speed (ω₀) is 0 rad/s.
Rearranging the equation, we get:
θ = (1/2)αt²
We can find the angular acceleration (α) using the formula:
ω = ω₀ + αt
Given:
ω = 12.0 rad/s
ω₀ = 0 rad/s
t = 3.00 s
Rearranging the equation, we get:
α = (ω - ω₀) / t
Substituting the given values:
α = (12.0 rad/s - 0 rad/s) / 3.00 s
α = 4.00 rad/s²
Now we can substitute the value of α into the equation for angular displacement:
θ = (1/2) * 4.00 rad/s² * (3.00 s)²
θ = (1/2) * 4.00 rad/s² * 9.00 s²
θ = 18.0 rad
Therefore, the wheel rotates by an angle of 18.0 radians in 3.00 seconds.
To find the angle (in radians) through which the wheel rotates, we can use the equations of rotational motion.
We'll use the equation:
θ = ω₀t + (1/2)αt²
where:
θ is the angle (in radians) through which the wheel rotates
ω₀ is the initial angular velocity (in rad/s)
α is the angular acceleration (in rad/s²)
t is the time (in seconds)
Given:
ω₀ = 0 rad/s (since the wheel starts from rest)
ω = 12.0 rad/s
t = 3.00 s
To find α, we can use the equation:
ω = ω₀ + αt
Substituting the given values:
12.0 rad/s = 0 rad/s + α(3.00 s)
Simplifying the equation, we get:
12.0 rad/s = 3.00 s α
Solving for α, we find:
α = 12.0 rad/s / 3.00 s
α = 4.00 rad/s²
Now we can substitute the values of α, ω₀, and t into the equation for θ:
θ = (0 rad/s)(3.00 s) + (1/2)(4.00 rad/s²)(3.00 s)²
Simplifying the equation, we get:
θ = (0 rad) + (1/2)(4.00 rad/s²)(9.00 s²)
θ = 18.00 rad
Therefore, the angle through which the wheel rotates in this time is 18.00 radians.