It is desired that the outer edee of a grindine wheel 9.0 cm in radius move at a rate of 6.0 m/s. (a) Determine the angular speed of the wheel. (b) What length of thread could be wound on the rim of the wheel in 3.0s when it is turning at this rate
(a) The linear velocity of a point on the edge of the wheel is given by v = rω, where r is the radius and ω is the angular speed. Rearranging this equation, we get ω = v/r.
Substituting the given values, we get:
ω = 6.0 m/s / 0.09 m = 66.7 rad/s
Therefore, the angular speed of the wheel is 66.7 rad/s.
(b) The length of thread wound on the rim of the wheel in 3.0 s is given by L = rθ, where r is the radius and θ is the angular displacement of the wheel in radians, which is equal to ωt.
Substituting the given values, we get:
θ = ωt = 66.7 rad/s × 3.0 s = 200.1 rad
L = 0.09 m × 200.1 rad = 18.01 m
Therefore, the length of thread that could be wound on the rim of the wheel in 3.0 s when it is turning at this rate is 18.01 m.
A force of 200 N acts tangentially on the rim of a wheel 25 cm in radius. (a) Find the torque. (b) Repeat if
the force makes an angle of 40 to a spoke of the wheel.
(a) The torque τ produced by the tangential force F at a distance r from the center of the wheel is given by τ = Fr. Substituting the given values, we get:
τ = 200 N × 0.25 m = 50 Nm
Therefore, the torque produced by the force is 50 Nm.
(b) If the force makes an angle of 40° to a spoke of the wheel, then the lever arm is the perpendicular distance from the line of action of the force to the axis of rotation, which is given by r sin θ, where r is the radius of the wheel and θ is the angle between the force and the spoke.
Substituting the given values, we get:
r sin θ = 0.25 m × sin 40° ≈ 0.16 m
The torque is now given by τ = Fr sin θ. Substituting the given values, we get:
τ = 200 N × 0.16 m ≈ 32 Nm
Therefore, the torque produced by the force at an angle of 40° to a spoke of the wheel is approximately 32 Nm.
To determine the angular speed of the wheel, use the formula:
Angular speed (ω) = Linear speed (v) / Radius (r)
Given:
Linear speed (v) = 6.0 m/s
Radius (r) = 9.0 cm = 0.09 m
(a) Calculating angular speed:
ω = v / r
ω = 6.0 m/s / 0.09 m
ω ≈ 66.67 rad/s
Therefore, the angular speed of the wheel is approximately 66.67 rad/s.
(b) To find the length of thread wound on the rim of the wheel in 3.0 s, we can use the formula:
Length of thread = Linear speed (v) × Time (t)
Given:
Linear speed (v) = 6.0 m/s
Time (t) = 3.0 s
Calculating length of thread:
Length of thread = v × t
Length of thread = 6.0 m/s × 3.0 s
Length of thread = 18.0 m
Therefore, when the wheel is turning at this rate, a length of 18.0 meters of thread could be wound on the rim in 3.0 seconds.