A wheel of mass 0.52kg and radius 46cm is spinning with an angular velocity of 23rad/s. You exert a force F on the wheel, the wheel comes to a stop after 1/4 turn. What is F.
To find the force exerted on the wheel, we need to use the law of conservation of angular momentum, which states that the initial angular momentum is equal to the final angular momentum.
The initial angular momentum (L_initial) is given by:
L_initial = I * ω_initial
where I is the moment of inertia of the wheel and ω_initial is the initial angular velocity.
Given:
Mass of the wheel (m) = 0.52 kg
Radius of the wheel (r) = 0.46 m
Angular velocity (ω_initial) = 23 rad/s
The moment of inertia of a solid disk rotating about its axis is given by:
I = (1/2) * m * r^2
Substituting the given values:
I = (1/2) * 0.52 kg * (0.46 m)^2
Now, we can calculate L_initial:
L_initial = I * ω_initial
To calculate the final angular momentum (L_final), we need to determine the final angular velocity (ω_final). We know that the wheel comes to a stop after 1/4 turn, which is equivalent to rotating by 90 degrees or π/2 radians.
The final angular velocity (ω_final) can be calculated using the formula:
ω_final = (2π) / (time taken to stop)
Since the wheel rotates 90 degrees (π/2 radians), and we know the initial angular velocity, we can find the time taken to stop using the formula:
θ = ω_initial * t
Solving for time (t):
π/2 = 23 rad/s * t
Now we can calculate ω_final:
ω_final = (2π) / t
Finally, we can calculate the final angular momentum (L_final):
L_final = I * ω_final
Since L_initial = L_final, we can set the two equations equal to each other:
I * ω_initial = I * ω_final
Now we can solve for the force (F) exerted on the wheel, as F is equal to the change in angular momentum divided by the time taken to stop:
F = ΔL / Δt
By rearranging the equation, we can solve for F:
F = (L_initial - L_final) / (time taken to stop)
Let me calculate the values and find the force (F) for you.