A 41.0 cm diameter wheel accelerates uniformly from 86.0 rpm to 342.0 rpm in 3.4 s. How far (in meters) will a point on the edge of the wheel have traveled in this time?

Circumference=pi*D = 3.14 * 41=128.8 cm=

1.288 m.

Vo = (86rev/min)*1min/60s*1.288m/rev =
1.85 m/s.
V = (342/86) * 1.85 = 7.34 m/s

a=(V-Vo)/t = (7.34-1.85)/3.4=1.61 m/s^2

d =Vo*t + 0.5*a*t^2
d=1.85*3.4 + 0.5*1.61*(3.4)^2= 15.6 m.

To find the distance traveled by a point on the edge of the wheel, we first need to calculate the angular acceleration of the wheel. We can then use this angular acceleration to find the angular displacement, and finally, use the diameter of the wheel to find the linear distance traveled.

Step 1: Calculate the angular acceleration.
Angular acceleration is the rate at which the angular velocity of the wheel changes.
Given:
Initial angular velocity, ω₁ = 86.0 rpm
Final angular velocity, ω₂ = 342.0 rpm
Time, t = 3.4 s

We can convert the angular velocities from rpm to rad/s:
ω₁ = (86.0 rpm) * (2π rad/1 min) * (1 min/60 s) = 9.03 rad/s
ω₂ = (342.0 rpm) * (2π rad/1 min) * (1 min/60 s) = 35.85 rad/s

The angular acceleration is then given by:
Angular acceleration, α = (ω₂ - ω₁) / t

Substituting the given values:
α = (35.85 rad/s - 9.03 rad/s) / 3.4 s
α = 26.82 rad/s / 3.4 s
α = 7.89 rad/s²

Step 2: Calculate the angular displacement.
Angular displacement, θ, is the change in the angle covered by the rotating wheel.
Using the kinematic equation for angular acceleration:
θ = ω₁ * t + 0.5 * α * t²

Substituting the values:
θ = (9.03 rad/s) * (3.4 s) + 0.5 * (7.89 rad/s²) * (3.4 s)²
θ ≈ 30.692 rad

Step 3: Calculate the distance traveled.
The distance traveled by a point on the edge of the wheel is the same as the circumference of the wheel, which is directly proportional to the angle covered (θ).
Given:
Diameter of the wheel, d = 41.0 cm

Circumference, C = π * d
C = π * (41.0 cm)
C ≈ 128.677 cm

To convert the circumference to meters:
C ≈ 128.677 cm * (1 m / 100 cm)
C ≈ 1.2877 m

To find the distance traveled by a point on the edge of the wheel, we multiply the circumference by the fraction of the angle covered.
Distance = C * (θ / 2π)
Distance ≈ 1.2877 m * (30.692 rad / 2π)
Distance ≈ 1.2877 m * (30.692 rad / 6.283 rad)
Distance ≈ 6.296 m

Therefore, a point on the edge of the wheel will have traveled approximately 6.296 meters in 3.4 seconds.