A 24- kg monkey climbs up a massless rope that runs over a frictionless tree limb and back down to a 35- kg package on the ground. What is the magnitude of the least acceleration the monkey must have if it is to lift the package off the ground?
m1=mass of monkey=24kg
m2=mass of package=35 kg
Minimum tension in the rope, T
= m2g
If the monkey has an acceleration of a, then tension produced, T
= m1(g+a)
Thus
m1(g+a) = m2*g
Solve for a.
To find the minimum acceleration required for the monkey to lift the package off the ground, we need to consider the forces acting on the system.
First, let's assign positive directions for the forces. Let's assume upward forces are positive and downward forces are negative.
For the monkey, the forces acting on it are:
1. The tension force in the rope pulling it upward.
2. The force of gravity pulling it downward (mg, where m is the mass of the monkey and g is the acceleration due to gravity).
For the package, the forces acting on it are:
1. The tension force in the rope pulling it upward.
2. The force of gravity pulling it downward (mg, where m is the mass of the package and g is the acceleration due to gravity).
Since the monkey and the package are connected by the same rope, the tension force in the rope is the same for both of them.
Now, let's write the equations for the forces acting on the monkey and the package:
For the monkey:
Tension - mg = ma₁, where a₁ is the acceleration of the monkey.
For the package:
Tension - mg = ma₂, where a₂ is the acceleration of the package.
To find the minimum acceleration, we need to find the maximum tension that can be achieved by the monkey. This happens when the monkey exerts the maximum force and the package exerts the minimum force.
The maximum force that the monkey can exert is its weight (mg), so the maximum tension is 2mg (the sum of the monkey's weight and the package's weight).
Substituting this value into the equations, we get:
2mg - mg = ma₁ (for the monkey)
2mg - mg = ma₂ (for the package)
Simplifying the equations:
mg = ma₁
mg = ma₂
We can cancel the masses:
g = a₁
g = a₂
Therefore, the minimum acceleration the monkey must have to lift the package off the ground is equal to the acceleration due to gravity, which is approximately 9.8 m/s².