A wheel with a weight of 395N comes off a moving truck and rolls without slipping along a highway. At the bottom of a hill it is rotating at an angular velocity of 23.0rad/s . The radius of the wheel is 0.607m and its moment of inertia about its rotation axis is 0.800 MR2. Friction does work on the wheel as it rolls up the hill to a stop, at a height of h above the bottom of the hill; this work has a magnitude of 3450J .

Calculate h. g is 9.81 which is the acceleration due to gravity.

Please help! I'm not sure how to answer this question.

45.3?

To calculate the height h, we need to consider the conservation of mechanical energy. At the bottom of the hill, the wheel has both kinetic energy due to its translational and rotational motion, and potential energy due to its height.

The initial kinetic energy (KE_initial) is given by the sum of the translational and rotational kinetic energies:

KE_initial = 1/2 * mv^2 + 1/2 * I * ω^2

where m is the mass of the wheel, v is its linear velocity, I is the moment of inertia, and ω is the angular velocity.

The final state is when the wheel comes to a stop at height h, so its final kinetic energy (KE_final) is zero.

The change in potential energy (ΔPE) is given by:

ΔPE = mgh

where g is the acceleration due to gravity.

According to the conservation of mechanical energy, the initial mechanical energy should be equal to the final mechanical energy:

KE_initial + ΔPE = KE_final

So we can write the equation as:

1/2 * mv^2 + 1/2 * I * ω^2 + mgh = 0

Now let's substitute the given values into the equation:

m = weight / g = 395N / 9.81 m/s^2
v = ? (initial linear velocity, unknown)
I = 0.800 * m * R^2 = 0.800 * 395N / 9.81 m/s^2 * (0.607m)^2
ω = 23.0 rad/s

We need to solve for v to find the initial linear velocity. Rearranging the equation, we get:

v^2 = (2 * (KE_final - KE_initial)) / m

substituting KE_initial and KE_final, and rearranging again, we have:

v^2 = (2 * (m * g * h - 1/2 * I * ω^2)) / m

Simplifying,

v^2 = 2 * g * h - 2 * I * ω^2 / m

Now we can substitute the values and solve for v:

v^2 = 2 * (9.81 m/s^2) * h - 2 * (0.800 * 395N / 9.81 m/s^2 * (0.607m)^2) * (23.0 rad/s)^2 / (395 N / 9.81 m/s^2)

After finding v, we can calculate h by rearranging the equation:

h = (v^2 * m) / (2 * g)

Substituting the known values will provide the answer for h.