A man seeking to set a world record wants to tow a 106,000-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 0.82. 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 to the airplane, we need to determine the maximum static friction force between his shoes and the runway. This maximum static friction force will limit the maximum force he can exert on the airplane, which in turn determines the maximum acceleration.
The formula for static friction is given by:
Static Friction Force = coefficient of static friction * Normal force
The normal force acting on the man is equal to his weight, since the weight acts perpendicular to the surface of the runway. Therefore, the normal force is:
Normal Force = mass of the man * acceleration due to gravity
Normal Force = 77 kg * 9.8 m/s^2
Normal Force = 754.6 N
Now we can calculate the maximum static friction force:
Static Friction Force = 0.82 * 754.6 N
Static Friction Force = 618.372 N
The maximum static friction force represents the maximum force the man can exert on the airplane. Since the mass of the airplane is given as 106,000 kg, we can calculate the maximum acceleration using Newton's second law:
Maximum force = mass of the airplane * maximum acceleration
618.372 N = 106,000 kg * maximum acceleration
Solving for maximum acceleration:
maximum acceleration = 618.372 N / 106,000 kg
maximum acceleration ≈ 0.00582 m/s^2
Therefore, the greatest acceleration the man can give the airplane is approximately 0.00582 m/s^2.