A belt passes over a wheel of radius 25 cm. If a point on the belt has a speed of 26 m/s. How fast is the wheel turning? Express your answer in rad/s.
To find the speed at which the wheel is turning, we need to use the relationship between linear speed and angular speed. The linear speed of a point on the belt is equal to the product of the angular speed of the wheel and the radius of the wheel.
Linear speed = angular speed × radius
Given:
Linear speed = 26 m/s
Radius of the wheel = 25 cm = 0.25 m
Substituting the given values into the equation, we have:
26 m/s = angular speed × 0.25 m
To solve for the angular speed (ω), divide both sides of the equation by 0.25 m:
angular speed = 26 m/s / 0.25 m
= 104 rad/s
Therefore, the wheel is turning at a speed of 104 rad/s.
To find the speed at which the wheel is turning, we can use the fact that the linear speed of a point on the belt is equal to the product of the angular speed of the wheel and the radius of the wheel.
Let's use the formula: v = ω * r
Where:
- v is the linear speed of a point on the belt (26 m/s).
- ω is the angular speed of the wheel (which we're trying to find).
- r is the radius of the wheel (25 cm or 0.25 m).
Rearranging the formula to solve for ω, we have: ω = v / r.
Plugging in the given values, we get: ω = 26 m/s / 0.25 m = 104 rad/s.
Therefore, the wheel is turning at a speed of 104 rad/s.
The circumference of the wheel is 50π cm = π/2 m
1 rotation covers π/2 m
x rotations cover 26 m
x/1 = 26/(π/2)
x = 52/π = appr 16.55
the wheel is turning at 16.55 rotations/second
or appr 993 rpm