A cord of negligible mass is wound around the rim of a flywheel of mass 20 kg and radius 20 cm. Steady pull of 25 N is applied in tue cord. The flywheel is mounted on a horizontal axis with frictionless bearings. Compute the angular acceleration of the wheel. Find the work done by the pull when 2m of the cord is unwounded. Find the kinetic energy of the wheel at this point.Given that M=20kg and R=20cm.
To find the angular acceleration of the flywheel, we can use the formula for torque:
τ = Iα
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
The torque τ is given by the product of the force applied and the radius of the flywheel:
τ = F × r
Plugging in the given values, we have:
τ = 25 N × 0.2 m
Next, we need to calculate the moment of inertia I. For a solid disk like the flywheel, the moment of inertia can be calculated using the formula:
I = (1/2)MR^2
Plugging in the given values, we have:
I = (1/2) × 20 kg × (0.2 m)^2
Now, we can plug in the values for τ and I into the equation for torque:
τ = Iα
25 N × 0.2 m = (1/2) × 20 kg × (0.2 m)^2 × α
Simplifying the equation, we can solve for α:
α = (25 N × 0.2 m) / ((1/2) × 20 kg × (0.2 m)^2)
= 2.5 rad/s^2
Therefore, the angular acceleration of the flywheel is 2.5 rad/s^2.
To find the work done by the pull when 2m of the cord is unwound, we can use the formula for work:
W = τθ
where W is the work done, τ is the torque, and θ is the angle through which the cord is unwound.
The torque τ is given by the force applied multiplied by the radius of the flywheel, as we calculated earlier.
To find θ, we can use the formula:
θ = s / r
where s is the length of the unwound cord, and r is the radius of the flywheel.
Plugging in the given values, we have:
s = 2 m
r = 0.2 m
Now we can calculate θ:
θ = 2 m / 0.2 m
= 10 radians
Finally, we can plug the values for τ and θ into the equation for work:
W = (25 N × 0.2 m) × 10 radians
= 50 Nm
Therefore, the work done by the pull when 2m of the cord is unwound is 50 Nm.
To find the kinetic energy of the wheel at this point, we can use the formula:
KE = (1/2) I ω^2
where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.
The angular velocity ω can be calculated from the angular acceleration α using the formula:
ω = α t
where t is the time taken for the cord to unwind.
Since the cord is unwounded at a steady rate, the time taken t can be calculated by dividing the length of the unwound cord by the rate of unwinding:
t = s / v
where v is the velocity of unwinding.
Since the velocity v is constant, we can find it using the formula:
v = ω r
Therefore:
t = s / (ω r)
Plugging in the values for s and r, we have:
t = 2 m / (ω × 0.2 m)
Now we can calculate ω using the given value for α:
ω = α t
= 2.5 rad/s^2 × (2 m / (2.5 rad/s^2 × 0.2 m)
= 20 rad/s
Next, using the given values for I and ω, we can calculate the kinetic energy:
KE = (1/2) × I × ω^2
= (1/2) × 20 kg × (0.2 m)^2 × (20 rad/s)^2
= 40 Joules
Therefore, the kinetic energy of the wheel when 2m of the cord is unwound is 40 Joules.