A solid cylinder is pivoted at its center about
a frictionless axle. A force is applied to the
outer radius of 1.27 m at an angle of 30 ◦
above the tangential and exerts a force of 5 N.
A second force is applied by wrapping rope
around the inner radius of 0.576 m, which
exerts a force of 8.36 N tangent to the cylinder.
The moment of inertia of the cylinder is
325 kg · m2
.
To determine the moment of inertia of the cylinder, we need to know some additional information about its mass distribution. The moment of inertia depends on how the mass of the cylinder is distributed relative to its axis of rotation.
The formula for the moment of inertia of a solid cylinder is given by the equation:
I = (1/2) * m * r^2,
where I is the moment of inertia, m is the mass of the cylinder, and r is the radius of the cylinder.
However, in this case, we are given the moment of inertia directly as 325 kg · m^2. Therefore, we don't need to calculate it using the above formula.
To find the torque exerted by the force applied at the outer radius, we can use the formula:
τ = F * r * sin(θ),
where τ is the torque, F is the force applied, r is the radius at which the force is applied, and θ is the angle between the force and the radius.
Given that F = 5 N and r = 1.27 m, we can substitute these values into the formula to find the torque:
τ1 = 5 N * 1.27 m * sin(30°).
To find the torque exerted by the force applied using the rope around the inner radius, we can use the same formula:
τ = F * r * sin(θ),
where F is the force applied, r is the radius at which the force is applied, and θ is the angle between the force and the radius.
Given that F = 8.36 N and r = 0.576 m, we can substitute these values into the formula to find the torque:
τ2 = 8.36 N * 0.576 m * sin(0°).
Finally, the total torque exerted on the cylinder is the sum of τ1 and τ2:
Total torque = τ1 + τ2.
Please note that in order to calculate the total torque in this example, we need to know the angle θ for both forces applied to the cylinder at the given radii. The angle θ is not provided in the given information, so we are unable to calculate the total torque without it.