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 solve the given problem, we can use the following equations:
(a) The moment of inertia of a hoop is given by the equation:
I = m * r^2
where I is the moment of inertia, m is the mass of the hoop, and r is the radius of the hoop.
Plugging in the values, we have:
I = 2.54 kg * (0.346 m)^2
Calculating this, we find:
I = 0.312 kg · m^2
Therefore, the moment of inertia of the tire about an axis passing through the hub at its center is 0.312 kg · m^2.
(b) The torque required to produce an angular acceleration can be calculated using the equation:
Torque = I * angular acceleration
Plugging in the values, we have:
Torque = 0.312 kg · m^2 * 0.710 rad/s^2
Calculating this, we find:
Torque = 0.221 N · m
Therefore, the torque required to produce an angular acceleration of 0.710 rad/s^2 is 0.221 N · m.
(c) To calculate the friction force required to create a torque, we use the equation:
Torque = friction force * radius
Rearranging the equation, we have:
friction force = Torque / radius
Plugging in the values, we have:
friction force = 0.221 N · m / 0.346 m
Calculating this, we find:
friction force ≈ 0.639 N
Therefore, the friction force applied tangentially to the edge of the tire that will create a torque of that magnitude is approximately 0.639 N.
To find the answers to these questions, we will need to use some formulas related to rotational motion and torque. Here's how you can solve each part:
(a) The moment of inertia of a hoop about an axis passing through its center is given by the formula:
I = m * r^2
where I is the moment of inertia, m is the mass of the hoop, and r is the radius of the hoop. Plugging in the given values:
I = 2.54 kg * (0.346 m)^2
Simplifying the equation will give you the answer in kg·m^2.
(b) The torque required to produce an angular acceleration is given by the formula:
τ = I * α
where τ is the torque, I is the moment of inertia, and α is the angular acceleration. Rearranging the formula, we get:
τ = I * α
Plugging in the values:
τ = (2.54 kg * (0.346 m)^2) * 0.710 rad/s^2
This will give you the answer in N·m.
(c) To find the friction force that will create a torque of the given magnitude, we can use the formula:
τ = F * r
where τ is the torque, F is the force, and r is the radius. Rearranging the formula, we get:
F = τ / r
Plugging in the values:
F = (0.710 N·m) / (0.346 m)
This will give you the answer in Newtons.