A rotating flywheel of diameter 48.0 cm uniformly accelerates from rest to 240 rad/s in 15.0 s. Find the linear velocity of a point on the rim of the wheel after 15.0 s. Round answer to one decimal place.
Well, we can calculate the linear velocity of a point on the rim of the wheel using the formula:
Linear velocity = (angular velocity) x (radius)
The angular velocity is given as 240 rad/s, and the radius of the wheel is half of its diameter, which is 48.0 cm / 2 = 24.0 cm.
So, let's do some unit conversions to make sure everything matches up. The radius needs to be converted to meters, so we have 24.0 cm = 0.24 m.
Now, we can substitute the values into the formula:
Linear velocity = (240 rad/s) x (0.24 m)
Linear velocity = 57.6 m/s
So, after 15.0 s, the linear velocity of a point on the rim of the wheel is approximately 57.6 m/s.
Now, excuse me while I go spin around in circles at that speed. Wheee!
To find the linear velocity of a point on the rim of the wheel after 15.0 seconds, we can use the formula:
Linear velocity = Angular velocity * Radius
First, let's find the angular velocity.
Given:
Initial angular velocity (ω₁) = 0 rad/s
Final angular velocity (ω₂) = 240 rad/s
Time (t) = 15.0 s
The angular acceleration (α) can be determined using the formula:
Angular acceleration (α) = (ω₂ - ω₁) / t
Substituting the given values:
α = (240 rad/s - 0 rad/s) / 15.0 s
α = 240 rad/s / 15.0 s
α ≈ 16.0 rad/s²
Next, let's find the radius of the flywheel.
Given:
Diameter (D) = 48.0 cm
The radius (r) can be calculated as:
r = D / 2
Substituting the given values:
r = 48.0 cm / 2
r = 24.0 cm
Now, let's calculate the angular velocity (ω) at the end of 15.0 seconds using the formula:
Angular velocity (ω₂) = Angular velocity (ω₁) + Angular acceleration (α) * Time (t)
Substituting the calculated values:
ω₂ = 0 rad/s + 16.0 rad/s² * 15.0 s
ω₂ = 240 rad/s
Finally, we can find the linear velocity (v) using the formula:
Linear velocity (v) = Angular velocity (ω₂) * Radius (r)
Substituting the calculated values:
v = 240 rad/s * 24.0 cm
v ≈ 5760 cm/s
Rounding to one decimal place, the linear velocity of a point on the rim of the wheel after 15.0 seconds is approximately 5760 cm/s.
To find the linear velocity of a point on the rim of the wheel, we first need to find the angular velocity of the wheel.
The formula to calculate angular velocity is:
Angular velocity (ω) = Change in angular displacement (Δθ) / Time taken (t)
Given:
Initial angular velocity (ω₀) = 0 rad/s (since the wheel starts from rest)
Final angular velocity (ω) = 240 rad/s
Time taken (t) = 15.0 s
Using the formula, we can calculate the change in angular displacement:
Δθ = ω - ω₀
Substituting the given values:
Δθ = 240 rad/s - 0 rad/s = 240 rad/s
Now we can find the angular velocity:
ω = Δθ / t
Substituting the known values:
ω = 240 rad/s / 15.0 s = 16 rad/s
The linear velocity of a point on the rim of the wheel can be calculated using the formula:
Linear velocity (v) = Radius (r) × Angular velocity (ω)
The radius of the wheel can be calculated as half of the diameter:
Radius (r) = Diameter (d) / 2 = 48.0 cm / 2 = 24.0 cm
Converting the radius to meters:
Radius (r) = 24.0 cm × (1 m / 100 cm) = 0.24 m
Substituting the values, we get:
Linear velocity (v) = 0.24 m × 16 rad/s = 3.84 m/s
Therefore, after 15.0 s, the linear velocity of a point on the rim of the wheel is 3.84 m/s.
The acceleration is centripetal (circular):
a = velocity^2 /radius (formula)
= 240^2 /24
= 2400 rad/s^2
Linear velocity is calculated using the formula v = u + at. That is:
final velocity = initial velocity + acceleration x time
linear velocity = 0 (since at rest) + 2400 x 15
= 36000 rad/s
I'm pretty sure this is what you do, although that final answer is quite large. I'd double check with someone else to be sure.