A man seeking to set a world record wants to tow a 106000-kg airplane along a runway by pulling horizontally on a cable attached to the airplane. The mass of the man is 77 kg, and the coefficient of static friction between his shoes and the runway is 0.99. What is the greatest acceleration the man can give the airplane? Assume that the airplane is on wheels that turn without any frictional resistance.
To find the greatest acceleration the man can give the airplane, we first need to determine the maximum force of static friction between the man's shoes and the runway.
The maximum force of static friction can be calculated using the formula:
Maximum force of static friction (fs) = coefficient of static friction (μs) * normal force (N)
The normal force (N) can be found by multiplying the mass of the man (m) by the acceleration due to gravity (g):
Normal force (N) = m * g
Given:
Mass of the man (m) = 77 kg
Coefficient of static friction (μs) = 0.99
Acceleration due to gravity (g) = 9.8 m/s^2
Plugging the values into the formula:
N = m * g
N = 77 kg * 9.8 m/s^2
N ≈ 754.6 N
fs = μs * N
fs = 0.99 * 754.6 N
fs ≈ 747.09 N
Now, we can calculate the maximum acceleration (a) that the man can give the airplane using Newton's second law of motion:
Force (F) = mass (m) * acceleration (a)
Rearranging the equation, we get:
a = F / m
Since the force is limited by the maximum force of static friction (fs), we have:
a = fs / m
Plugging in the values:
a = 747.09 N / 106000 kg
a ≈ 0.0070 m/s^2
Therefore, the greatest acceleration the man can give the airplane is approximately 0.0070 m/s^2.