To hoist himself into a tree, a 71.0-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 367 N. Neglect any friction between the rope and the branch, and determine the man's upward acceleration.
Use
(net Force) = ma
and figure out the upward force F applied by the rope on the climber's hands. The NET force on the climber is more than F. Can you see why?
5.17
To determine the man's upward acceleration, we need to apply 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 (F = ma).
First, let's calculate the force that results from the man's weight. The weight can be calculated by multiplying the mass of the man (71.0 kg) by the acceleration due to gravity (9.8 m/s^2):
Weight = mass x acceleration due to gravity
Weight = 71.0 kg x 9.8 m/s^2
Weight = 695.8 N
Next, we need to determine the net force acting on the man. The net force is the difference between the force he exerts on the rope and his weight:
Net force = Force exerted by the man - Weight
Net force = 367 N - 695.8 N
Net force = -328.8 N
The negative sign indicates that the net force is downward, as it opposes the upward motion of the man.
Now, we can use Newton's second law to find the man's upward acceleration:
Net force = mass x acceleration
-328.8 N = 71.0 kg x acceleration
Rearranging the equation to solve for acceleration:
acceleration = Net force / mass
acceleration = -328.8 N / 71.0 kg
acceleration ≈ -4.63 m/s^2
The negative sign indicates that the man's acceleration is in the opposite direction of his force exertion. Therefore, the man's upward acceleration is approximately -4.63 m/s^2.