To hoist himself into a tree, a 66.0-kg man ties one end of a nylon rope around his waist and throws the other end over a branch of the tree. He then pulls downward on the free end of the rope with a force of 366 N. Neglect any friction between the rope and the branch, and determine the man's upward acceleration.
F = ma,
a = F/m = 366 N / 66 kg = 5.5 m/s^2.
To determine the man's upward acceleration, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.
Given:
Mass of the man (m) = 66.0 kg
Force exerted by the man (F) = 366 N
The force exerted by the man is in the downward direction, so we'll consider its magnitude as negative. The net force acting on the man is given by:
Net force (F_net) = -F
Using Newton's second law, we have:
F_net = m * a
-366 N = 66.0 kg * a
We can rearrange the equation to solve for the upward acceleration (a):
a = (-366 N) / (66.0 kg)
a ≈ -5.55 m/s²
The negative sign indicates that the acceleration is in the upward direction, opposing gravity. Thus, the man's upward acceleration is approximately 5.55 m/s².
To determine the man's upward acceleration, we can use Newton's second law of motion, which states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration. In this case, the force exerted on the man is the tension in the rope.
First, let's calculate the tension in the rope. The tension in the rope is equal to the force the man exerts on the free end of the rope.
Given:
- Mass of the man (m) = 66.0 kg
- Force exerted by the man (F) = 366 N
Therefore, tension in the rope (T) = Force exerted by the man.
T = F
T = 366 N
Since there is no friction between the rope and the branch, the tension in the rope will be equal to the man's weight (mg), which acts as the force causing the upward acceleration.
T = mg
Substituting the values:
366 N = 66.0 kg * g
To solve for the acceleration, we need to find the gravitational acceleration (g). The standard value for gravitational acceleration on Earth is approximately 9.8 m/s^2.
366 N = 66.0 kg * 9.8 m/s^2
Now we can solve for the acceleration:
366 N = 646.8 kg·m/s^2
Divide both sides by the mass:
a = 646.8 kg·m/s^2 / 66.0 kg
a ≈ 9.8 m/s^2
The man's upward acceleration is approximately 9.8 m/s^2.