Torque Question airplane wheels

[wannabenerd]

1. An airliner lands with a speed of 45.0 m/s. Each wheel of the plane has a radius of 1.25 m and a moment of inertia of 110 kg·m2. At touchdown the wheels begin to spin under the action of friction. Each wheel supports a weight of 1.40 x 10^4 N, and the wheels attain the angular speed of rolling without slipping in 0.53 s. What is the coefficient of kinetic friction between the wheels and the runway? Assume that the speed of the plane is constant.



2. Homework Equations :
v=rw
I= 2/5MR^2
F=ma
friction=un
sum of forces?


3. I started by finding the angular velocity. using v=rw, i found it to be 36 rad/s. Then, I found the angular acceleration by doing w=w+at and a is equal to 67.925 rad/s^2. I don't know what to do next? Should I do the F=ma? But a is the linear acceleration and I don't know. I'm really confused about this problem because friction acts on four wheels or something? I don't know. Help please?

 

[Office_Shredder]

The rotational version of F=ma is T=W*a_r where a_r is rotational acceleration, W is the moment of inertia, and T is the torque

 

[ogb p]

I would suggest breaking down the problem into smaller parts and using the relevant equations to solve for the coefficient of kinetic friction. Here are the steps I would suggest:

Step 1: Calculate the total weight supported by all four wheels.

To find the total weight, we can simply multiply the weight supported by one wheel (1.40 x 10^4 N) by the number of wheels (4). This gives us a total weight of 5.60 x 10^4 N.

Step 2: Calculate the net torque acting on the wheels.

Since the plane is landing with a constant speed, we can assume that the net force acting on the wheels is zero. Therefore, the only torque acting on the wheels is the frictional torque. We can calculate this torque by using the equation T = rF, where r is the radius of the wheel and F is the frictional force. The frictional force can be calculated using the equation F = μN, where μ is the coefficient of kinetic friction and N is the normal force (in this case, equal to the weight of the plane). Therefore, the net torque acting on the wheels can be written as T = μrN.

Step 3: Calculate the moment of inertia of the wheels.

The moment of inertia of each wheel can be calculated using the equation I = 2/5MR^2, where M is the mass of the wheel and R is the radius. In this case, we are given the moment of inertia (110 kg·m^2), so we can use the equation to solve for the mass of each wheel.

Step 4: Set up the equation for rotational motion.

Using the equation T = Iα, where T is the net torque, I is the moment of inertia, and α is the angular acceleration, we can set up the equation for rotational motion. In this case, we know the values for T and I, and we can solve for α.

Step 5: Calculate the angular acceleration.

Substituting the values into the equation from step 4, we get μrN = (2/5)MR^2α. Rearranging the equation, we get α = (5μN)/(2MR).

Step 6: Calculate the time for the wheels to start rolling without slipping.

Using the equation ω = ω0 + αt, where ω is the final angular velocity, ω0