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 need to consider the forces acting on the system.
The force of friction between the man's shoes and the runway opposes the motion and allows the man to move the airplane. The maximum force of static friction (Fstatic) can be calculated using the equation:
Fstatic = μs * N
where μs is the coefficient of static friction and N is the normal force.
The normal force (N) can be calculated as the weight of the man and the airplane combined, since they are on the ground:
N = (mass of man + mass of airplane) * g
where g is the acceleration due to gravity, approximately 9.8 m/s^2.
N = (77 kg + 106000 kg) * 9.8 m/s^2
N ≈ 1,045,060 N
Now we can calculate the maximum force of static friction using the coefficient of static friction:
Fstatic = 0.99 * 1,045,060 N
Fstatic ≈ 1,035,599.4 N
Since force (F) is equal to mass (m) multiplied by acceleration (a), we can rearrange the equation to find the maximum acceleration (a):
a = Fstatic / (mass of man + mass of airplane)
a ≈ 1,035,599.4 N / (77 kg + 106000 kg)
a ≈ 1,035,599.4 N / 106,077 kg
a ≈ 9.76 m/s^2
Therefore, the greatest acceleration the man can give the airplane is approximately 9.76 m/s^2.