A bicycle tire has a mass of 2.54 kg and a radius of 0.346 m.
(a) Treating the tire as a hoop, what is its moment of inertia about an axis passing through the hub at its center?
kg · m2
(b) What torque is required to produce an angular acceleration of 0.710 rad/s2?
N · m
(c) What friction force applied tangentially to the edge of the tire will create a torque of that magnitude?
N
To find the answers to these questions, we need to understand the concepts of moment of inertia, torque, and friction. Let's break down each question and explain how to find the answers.
(a) Moment of inertia (I) is a measure of an object's resistance to changes in its rotation. For a hoop, the moment of inertia can be calculated using the formula: I = m * r^2, where m is the mass of the object and r is the radius. In this case, the mass of the bicycle tire is given as 2.54 kg and the radius is given as 0.346 m. Plugging these values into the formula, we can calculate the moment of inertia.
I = (2.54 kg) * (0.346 m)^2
Using this formula, you can find the moment of inertia of the tire in kg · m^2.
(b) Torque (τ) is a measure of the force required to rotate an object. It can be calculated using the formula: τ = I * α, where I is the moment of inertia and α is the angular acceleration. In this case, the angular acceleration is given as 0.710 rad/s^2. We already calculated the moment of inertia in part (a). Plugging these values into the formula, we can find the torque required.
τ = (moment of inertia) * (angular acceleration)
Using this formula, you can find the torque required to produce the given angular acceleration in N · m.
(c) Friction force (F) is the force that resists the relative motion of objects in contact. In this case, we want to find the friction force that will create a torque of the same magnitude as in part (b). To do this, we need to consider the relationship between torque and force. Torque (τ) can be calculated using the formula: τ = r * F, where r is the radius and F is the force. Rearranging the formula, we can solve for force:
F = τ / r
In part (b), we calculated the torque required. Additionally, the radius of the tire is given as 0.346 m. Plugging these values into the formula, we can calculate the friction force required.
F = (torque) / (radius)
Using this formula, you can find the friction force required in N.