A wheel 32.6 cm in diameter accelerates uniformly from 247 rpm to 378 rpm in 6.54 s. How far will a point on the edge of the wheel have traveled in this time
Circumference = pi*Dia = 3.14 * 32.6cm = 102.4 cm = 1.024 m.
Vo = 247rev/60s. * 1.024m/rev=4.22 m/s.
V = 378rev/60s * 1.024m/rev = 6.45 m/s.
V = Vo + a*t
V = 6.45 m/s
Vo = 4.22 m/s
t = 6.54 s.
Solve for a.
d = Vo*t + 0.5a*t^2
To find the distance traveled by a point on the edge of the wheel, we need to determine the circumference of the wheel and then multiply it by the number of revolutions.
1. First, let's calculate the initial and final angular velocities (ω1 and ω2) in radians per second. We have the initial and final angular velocities in rotations per minute (rpm), so we need to convert them.
Given:
- Initial angular velocity (ω1) = 247 rpm
- Final angular velocity (ω2) = 378 rpm
To convert rpm to radians per second, we use the conversion factor: 1 rpm = π/30 radians per second.
So,
ω1 = 247 rpm × π/30 radians per second
ω2 = 378 rpm × π/30 radians per second
2. Next, let's calculate the angular acceleration (α) using the formula:
α = (ω2 - ω1) / t
Where:
- ω1 is the initial angular velocity in radians per second
- ω2 is the final angular velocity in radians per second
- t is the time in seconds
Given values:
- ω1 (initial angular velocity)
- ω2 (final angular velocity)
- t (time) = 6.54 s
Plug in the values and calculate α.
3. Once we have the angular acceleration (α), we can determine the distance traveled using the formula for uniform angular acceleration:
θ = ω1*t + 1/2 * α * t^2
Where:
- θ is the angular displacement in radians
- ω1 is the initial angular velocity in radians per second
- α is the angular acceleration in radians per second squared
- t is the time in seconds
4. Finally, we can calculate the distance traveled by multiplying the angular displacement (θ) by the circumference of the wheel.
The circumference (C) of a wheel can be calculated using the formula:
C = 2 * π * r
Where:
- C is the circumference of the wheel
- r is the radius of the wheel, which is half of the diameter
Given value:
- Diameter of the wheel = 32.6 cm
Convert the diameter to radius by dividing it by 2.
Finally, multiply the angular displacement (θ) by the circumference (C) to find the distance traveled by a point on the edge of the wheel.