Question

An automobile tire is shown in the figure below. The tire is made of rubber with a uniform density of 1.10 ✕ 103 kg/m3. The tire can be modeled as consisting of two flat sidewalls and a tread region. Each of the sidewalls has an inner radius of 16.5 cm and an outer radius of 30.5 cm as shown, and a uniform thickness of 0.655 cm. The tread region can be approximated as having a uniform thickness of 2.50 cm (that is, its inner radius is 30.5 cm and outer radius is 33.0 cm as shown) and a width of 20.2 cm.

What is the moment of inertia (in kg · m2) of the tire about an axis perpendicular to the page through its center?
kg · m2

Answer 1

The moment of inertia of the tire about an axis perpendicular to the page through its center is 0.219 kg · m².

Step-by-step explanation:

To determine the moment of inertia of the tire, we can calculate the contributions from the sidewalls and the tread region separately, then sum them up. For the sidewalls, the moment of inertia can be calculated using the formula for the moment of inertia of a thin-walled cylindrical shell: I = ½m(r_outer² + r_inner²), where m is the mass and r_outer and r_inner are the outer and inner radii, respectively. For each sidewall, the mass can be found by multiplying the density by the volume (π * thickness * (r_outer² - r_inner²)), resulting in a moment of inertia for one sidewall of 0.0397 kg · m². Since there are two sidewalls, their combined moment of inertia is 2 * 0.0397 = 0.0794 kg · m².

Moving to the tread region, the moment of inertia of a solid cylinder is given by I = ½m(r_outer² + r_inner²). Calculating the mass of the tread region (π * width * thickness * (r_outer² - r_inner²)) and applying the formula yields a moment of inertia for the tread region of 0.1396 kg · m^2.

Adding the contributions from the sidewalls and the tread region together, we get 0.0794 + 0.1396 = 0.219 kg · m² as the final moment of inertia of the tire about the specified axis.

Answer 2

To find the moment of inertia of the tire, calculate the moment of inertia of each component separately and then add them up.

Step-by-step explanation:

To find the moment of inertia of the tire, we need to calculate the moment of inertia of each component separately and then add them up.

For the sidewalls, we can use the formula for the moment of inertia of a hollow cylinder: I = (1/2) * M * (R2 + r2), where M is the mass, R is the outer radius, and r is the inner radius.

For the tread region, we can use the formula for the moment of inertia of a solid cylinder: I = (1/2) * M * R2, where M is the mass and R is the radius.

Finally, we can add these two values to get the total moment of inertia of the tire.