A bicycle has wheels with a diameter of 0.630 m. It accelerates uniformly and the rate of rotation of its wheels increases from 197 rpm to 300 rpm in a time of 22.5 s. Find the linear acceleration of the bicycle
R=v₀•t
t=R/v₀
197 rpm = 197/60 =3.28 rev/s
300 = 5 rev/s
ω=ω₀+εt
ε= (ω-ω₀)/t =(5-3.28)/22.5=0.0076 rad/s
a = ε•R=0.0076•0.63=0.0048 m/s²
you forgot to convert form from frequency to angular velocity where
197rpm=3.28rev/s*2pi
apart from that the steps are accurate
To find the linear acceleration of the bicycle, we can use the following formula:
linear acceleration = (final angular velocity - initial angular velocity) * radius
Before applying the formula, we need to convert the angular velocities from revolutions per minute (rpm) to radians per second (rad/s).
1 revolution = 2π radians
So, the formula becomes:
linear acceleration = ((final angular velocity in rad/s) - (initial angular velocity in rad/s)) * radius
Given:
- Initial angular velocity (ω1) = 197 rpm
- Final angular velocity (ω2) = 300 rpm
- Time (t) = 22.5 s
- Diameter of the wheels (d) = 0.630 m
First, let's convert the angular velocities to rad/s:
ω1 = 197 rpm = 197 * 2π rad/min = (197 * 2π) / 60 rad/s ≈ 20.68 rad/s
ω2 = 300 rpm = 300 * 2π rad/min = (300 * 2π) / 60 rad/s ≈ 31.42 rad/s
Next, let's calculate the radius:
radius (r) = diameter / 2 = 0.630 m / 2 = 0.315 m
Now, substitute the values into the formula:
linear acceleration = (ω2 - ω1) * radius
= (31.42 rad/s - 20.68 rad/s) * 0.315 m
= 10.74 rad/s * 0.315 m
≈ 3.38 m/s²
Therefore, the linear acceleration of the bicycle is approximately 3.38 m/s².
To find the linear acceleration of the bicycle, we need to convert the rate of rotation from rpm (revolutions per minute) to radians per second.
1 revolution = 2π radians
First, let's convert the initial and final rates of rotation from rpm to radians per second:
Initial rate of rotation: 197 rpm
Final rate of rotation: 300 rpm
To convert rpm to radians per second, we can use the following conversion factor:
1 rpm = (2π/60) radians per second
Initial rate of rotation in radians per second:
(197 rpm) * (2π/60) = 20.7 rad/s
Final rate of rotation in radians per second:
(300 rpm) * (2π/60) = 31.4 rad/s
Next, we can calculate the change in rate of rotation by subtracting the initial rate from the final rate:
Change in rate of rotation = Final rate of rotation - Initial rate of rotation
Change in rate of rotation = 31.4 rad/s - 20.7 rad/s
Change in rate of rotation = 10.7 rad/s
Now, we can use the linear acceleration formula to find the acceleration of the bicycle:
Linear Acceleration = Δω / Δt
where Δω is the change in rate of rotation and Δt is the time interval.
Linear Acceleration = 10.7 rad/s / 22.5 s
Calculating this value, we get:
Linear Acceleration = 0.475 m/s²
Therefore, the linear acceleration of the bicycle is 0.475 m/s².