An airliner lands with a speed of 48.5 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 ✕ 104 N, and the wheels attain their angular speed in 0.432 s and start rolling without slipping. What is the coefficient of friction between the wheels and the runway? Assume that the speed of the plane is constant
To find the coefficient of friction between the wheels and the runway, we can use the following steps:
Step 1: Calculate the linear speed of the wheels
The linear speed of the wheels can be calculated using the formula:
v = ω * r
Where:
v = linear speed
ω = angular speed
r = radius of the wheel
Given:
ω = unknown
r = 1.25 m
We know that the linear speed of the wheels is equal to the speed of the airliner, which is 48.5 m/s. Therefore, we can set up the equation:
48.5 m/s = ω * 1.25 m
Dividing both sides by 1.25 m, we find:
ω = 48.5 m/s / 1.25 m
ω = 38.8 rad/s
Step 2: Calculate the acceleration of the wheels
The acceleration of the wheels can be calculated using the following equation:
α = ω / t
Where:
α = angular acceleration
ω = angular speed
t = time taken to reach the final angular speed
Given:
ω = 38.8 rad/s
t = 0.432 s
We can substitute the values into the equation:
α = 38.8 rad/s / 0.432 s
α ≈ 89.81 rad/s^2
Step 3: Calculate the net torque on the wheels
To calculate the net torque on the wheels, we need to use the equation:
τ = I * α
Where:
τ = torque
I = moment of inertia
α = angular acceleration
Given:
I = 110 kg · m^2
α ≈ 89.81 rad/s^2
Substituting the values into the equation:
τ = 110 kg · m^2 * 89.81 rad/s^2
τ ≈ 9.8791 * 10^3 N · m
Step 4: Calculate the normal force on each wheel
The normal force on each wheel can be calculated using the equation:
N = m * g
Where:
N = normal force
m = mass
g = acceleration due to gravity
Given:
N = 1.40 * 10^4 N
To find the mass, we can use the equation:
mg = N
m = N / g
Substituting the values into the equation:
m = (1.40 * 10^4 N) / (9.8 m/s^2)
m ≈ 1.43 * 10^3 kg
Step 5: Calculate the force of friction
The force of friction can be calculated using the equation:
f = μ * N
Where:
f = force of friction
μ = coefficient of friction
N = normal force
Given:
N = 1.40 * 10^4 N
Substituting the values into the equation:
f = μ * 1.40 * 10^4 N
Step 6: Calculate the torque due to friction
The torque due to friction can be calculated using the equation:
τ = f * r
Where:
τ = torque
f = force of friction
r = radius of the wheel
Given:
r = 1.25 m
Substituting the values into the equation:
τ = f * 1.25 m
Step 7: Equate torques to find coefficient of friction
The net torque on the wheels due to friction should be equal to the torque generated by the air resistance. Therefore, we can equate the two torques:
τ = I * α = f * r
Given:
τ ≈ 9.8791 * 10^3 N · m
r = 1.25 m
Substituting the values into the equation:
9.8791 * 10^3 N · m = f * 1.25 m
Dividing both sides by 1.25 m, we find:
f = (9.8791 * 10^3 N · m) / 1.25 m
f ≈ 7.903 * 10^3 N
Finally, substituting the value of force of friction (f) into the equation for the coefficient of friction:
μ = f / N ≈ (7.903 * 10^3 N) / (1.40 * 10^4 N)
μ ≈ 0.564
Therefore, the coefficient of friction between the wheels and the runway is approximately 0.564.
To find the coefficient of friction between the wheels and the runway, we need to use the principles of rotational dynamics.
1. Calculate the initial angular speed (ω_i) of the wheels:
We are given the time it takes for the wheels to attain their angular speed (0.432 s) and the final angular speed (ω_f = 0 rad/s since the wheels start rolling without slipping).
Since angular acceleration (α) is constant, we can use the equation:
ω_f = ω_i + αt
Rearrange the equation to solve for ω_i:
ω_i = ω_f - αt
Since ω_f = 0 rad/s:
ω_i = 0 - αt
2. Calculate the angular acceleration (α):
We can use the equation relating linear acceleration (a) and angular acceleration (α):
a = αr
Rearrange the equation to solve for α:
α = a / r
The linear acceleration (a) can be calculated using the equation:
a = Δv / t
Δv is the change in velocity, which is equal to the final velocity (v_f = 48.5 m/s) since the speed of the plane is constant.
Rearrange the equation to solve for a:
a = v_f / t
3. Calculate the normal force (N) acting on each wheel:
The weight of the plane is equal to the normal force acting on each wheel.
N = mg
m is the mass of the plane, which we can calculate using the weight given (1.40 × 10^4 N) and the acceleration due to gravity (g ≈ 9.8 m/s^2).
Rearrange the equation to solve for m:
m = N / g
4. Calculate the frictional force (f):
The frictional force is equal to the product of the coefficient of friction (μ) and the normal force (N).
f = μN
5. Calculate the torque (τ) exerted on each wheel due to the frictional force:
Torque is given by the equation:
τ = Iα
Rearrange the equation to solve for α:
α = τ / I
6. Equate the torque due to friction (τ) with the force of friction (f) multiplied by the radius (r) of the wheel:
τ = f * r
Substituting the values of τ and α, we get:
f * r = I * α
7. Solve for the coefficient of friction (μ):
Rearrange the equation to solve for μ:
μ = (I * α) / (N * r)
Substitute the values of I, α, N, and r calculated above.
Plug in the given values and follow the steps to calculate the coefficient of friction.