A man seeking to set a world record wants to tow a 104,000-kg airplane along a runway by pulling horizontally on a cable attached to the airplane. The mass of the man is 93 kg, and the coefficient of static friction between his shoes and the runway is 0.78. What is the greatest acceleration the man can give the airplane? Assume that the airplane is on wheels that turn without any frictional resistance.
Max force the man can apply is
F= F(fr)= μN=μmg=0.78•93•9.8=710.9 N
Max acceleration
a=F/M = 710.9/104000=0.0068 m/s²
To find the greatest acceleration the man can give the airplane, we need to consider the forces acting on the system.
The force of static friction between the man's shoes and the runway opposes the applied force of the man pulling the airplane. When this force reaches its maximum value, the airplane will start moving.
The formula for the force of static friction is:
\(F_{\text{friction}} = \mu_s \cdot F_{\text{normal}}\),
where \(\mu_s\) is the coefficient of static friction and \(F_{\text{normal}}\) is the normal force exerted on the man.
The normal force is equal to the weight of the man:
\(F_{\text{normal}} = m_{\text{man}} \cdot g\),
where \(m_{\text{man}}\) is the mass of the man and \(g\) is the acceleration due to gravity.
The maximum force of static friction can be found using the formula:
\(F_{\text{friction, max}} = \mu_s \cdot F_{\text{normal}}\).
Since the man is exerting the maximum force of static friction on the airplane, this force is equal to the force required to accelerate the airplane.
Using Newton's second law, \(F = m \cdot a\), where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration, we can set up the following equation:
\(F_{\text{friction, max}} = m_{\text{plane}} \cdot a\),
where \(m_{\text{plane}}\) is the mass of the plane.
Now we can substitute the relevant values into the equations:
\(F_{\text{normal}} = m_{\text{man}} \cdot g\),
\(F_{\text{friction, max}} = \mu_s \cdot F_{\text{normal}}\),
\(F_{\text{friction, max}} = m_{\text{plane}} \cdot a\).
First, calculate the normal force:
\(F_{\text{normal}} = m_{\text{man}} \cdot g = 93 \text{ kg} \cdot 9.8 \text{ m/s}^2\).
Next, calculate the maximum force of static friction:
\(F_{\text{friction, max}} = \mu_s \cdot F_{\text{normal}} = 0.78 \cdot F_{\text{normal}}\).
Finally, solve for the acceleration:
\(a = \frac{F_{\text{friction, max}}}{m_{\text{plane}}}\).
Substitute the known values into the equation and solve for the acceleration:
\(a = \frac{0.78 \cdot m_{\text{man}} \cdot g}{m_{\text{plane}}}\).
Using the given values, \(m_{\text{man}} = 93 \text{ kg}\), \(g = 9.8 \text{ m/s}^2\), and \(m_{\text{plane}} = 104,000 \text{ kg}\), we can calculate the acceleration \(a\).