A person thought it would be a good idea to change a tire on the side of a hill. The tire has a width

of W = 0.30 m and a radius of R = 0.40 m. The inside of the wheel is empty, but the wheel is solid
from a radius of R/2 to R with uniform density. The density of the material is 2000 kg/m3
.
(a) What is the mass, weight and moment of inertia of the wheel?
(b) If the wheel starts from rest and rolls down a 4.00 m ramp, 30.0
o
relative to the horizontal, what
is the wheel’s speed at the bottom of the ramp?
(c) What average torque do you need to apply to stop it at the bottom of the ramp in one revolution?

first mass of rubber

= 2000 * .3 * ∫ 2 pi r dr
from r = .2 to r = .4

= 600 (2 pi) (1/2)(.16-.04)

= 226 kg

226 * 9.81 = 2219 Newtons weight

I = 2000*.3 *∫ 2 pi r^3 dr
= 3770 (1/4)(.4^4-.2^4)
= 22.6
===================

change in potential energy = m g h

= 2219*4*sin 30 = 4438 Joules

= (1/2)mv^2 + (1/2)I w^2
but w = v/r = v/.4 = 2.5 v
so
4438 = (1/2)(226)v^2 + (1/2)(22.6)(6.25 v^2)
or
4438 = 184 v^2
so
v = 24.2 m/s
and w = 60.4 radians/s

Torque * angle = change in Ke
T * 2 pi = (1/2) I w^2

To solve this problem, we need to apply the principles of mechanics, specifically considering the mass, weight, moment of inertia, and rotational motion of the wheel.

(a) First, let's calculate the mass, weight, and moment of inertia of the wheel.

1. Mass of the Wheel:
To find the mass, we need to calculate the volume of the wheel and then multiply it by the density.

The volume of the wheel can be calculated by subtracting the volume of the hollow part from the volume of the entire wheel. The volume of the hollow part is a cylinder with a height equal to the width of the wheel (W) and a radius equal to R/2. The volume of the entire wheel is a cylinder with a height equal to the width of the wheel (W) and a radius equal to R.

Volume of the hollow part = π(R/2)^2 * W
Volume of the entire wheel = πR^2 * W

Now, subtracting the volume of the hollow part from the volume of the entire wheel, we get:

Volume of the wheel = πR^2 * W - π(R/2)^2 * W = πW(R^2 - (R/2)^2)

The mass of the wheel is given by:

Mass = Density * Volume = 2000 kg/m^3 * πW(R^2 - (R/2)^2)

2. Weight of the Wheel:
Weight is the force with which an object is pulled towards the center of the Earth. It can be calculated using the formula:

Weight = Mass * gravitational acceleration

The gravitational acceleration is approximately 9.8 m/s^2.

Weight = Mass * 9.8 m/s^2

3. Moment of Inertia:
Moment of inertia, for a solid cylinder rotating about its central axis, is given by:

I = (1/2) * Mass * R^2

(b) To find the wheel's speed at the bottom of the ramp, we need to apply the principle of conservation of energy.

The initial potential energy of the wheel is converted into the kinetic energy at the bottom of the ramp. Assuming no energy losses due to friction or other factors, we can calculate the speed at the bottom of the ramp using the equation:

mgh = (1/2) * mv^2

Where:
m = mass of the wheel
g = gravitational acceleration
h = height of the ramp
v = speed of the wheel at the bottom of the ramp

Since the wheel starts from rest, the initial velocity is zero (v₀ = 0). Solving the equation for v, we get:

v = sqrt(2gh)

(c) To find the average torque needed to stop the wheel at the bottom of the ramp in one revolution, we use the equation:

Torque = Moment of Inertia * Angular Acceleration

Since the wheel is rotating one revolution (360 degrees), the angular acceleration can be calculated using the equation:

Angular Acceleration = (Final Angular Velocity - Initial Angular Velocity) / Time

The initial angular velocity is 0 (ω₀ = 0). The final angular velocity can be calculated using the equation:

v = ω * R

Where:
v = linear velocity of the wheel at the bottom of the ramp
ω = angular velocity
R = radius of the wheel

The time can be calculated by dividing the angle (360 degrees) by the angular velocity:

Time = Angle / Angular Velocity

Substitute the values into the equation: Torque = Moment of Inertia * Angular Acceleration, to find the average torque needed to stop the wheel.

I hope this helps you solve the problem! Let me know if you have further questions.