An inventive child named Chris wants to reach an apple in a tree without climbing the tree. Sitting in a chair connected to a rope that passes over a frictionless pulley, Chris pulls on the loose end of the rope with such a force that the spring scale reads 265 N. Chris’s true weight is 287 N, and the chair weighs 223 N.
a) Find the magnitude of the acceleration of the system. The acceleration due to gravity is g = 9.8 m/s2 .
Answer in units of m/s2
b) Find the magnitude of the force Chris exerts on the chair.
Answer in units of N
Elena is correct for part a but for part b you need to subtract the tension for the answer.
N=mg+ma-T = 287 + 287•0.38/9.8 -265 = 33.13 N
To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In this case, the object is the combined system of Chris, the chair, and the apple.
a) To find the magnitude of the acceleration of the system, we need to calculate the net force acting on the system.
The forces acting on the system are the force Chris exerts on the rope (265 N), Chris's weight (287 N), and the weight of the chair (223 N). The forces due to gravity are acting downwards, while the force Chris exerts on the rope is acting upwards.
Let's denote the magnitude of the acceleration as "a" and set up the equation using Newton's second law:
Net force = mass * acceleration
The net force can be calculated as the sum of the forces:
Net force = Force Chris exerts on the rope - Chris's weight - Weight of the chair
Net force = 265 N - 287 N - 223 N
Once we have calculated the net force acting on the system, we can substitute it into the equation:
Net force = (265 N - 287 N - 223 N) = mass * acceleration
To solve for the magnitude of the acceleration, we rearrange the equation:
acceleration = Net force / mass
Now we can substitute the values into the equation:
acceleration = (-245 N) / (mass)
The mass of the system can be found by adding Chris's weight and the weight of the chair:
mass = Chris's weight + Weight of the chair
mass = 287 N + 223 N
Plugging in the values, we have:
acceleration = (-245 N) / (510 N)
acceleration ≈ -0.480 m/s^2
However, the acceleration cannot be negative in this case since it is a magnitude. Therefore, we take the absolute value:
acceleration ≈ 0.480 m/s^2
b) To find the magnitude of the force Chris exerts on the chair, we need to use the same equation as before:
Net force = mass * acceleration
The force Chris exerts on the chair can be calculated by rearranging the equation:
Force Chris exerts on the chair = mass * acceleration
Substituting the values, we have:
Force Chris exerts on the chair = (510 N) * (0.480 m/s^2)
Force Chris exerts on the chair ≈ 245 N
Therefore, the magnitude of the force Chris exerts on the chair is approximately 245 N.
To find the magnitude of the acceleration of the system, we first need to calculate the net force acting on the system.
The weight of Chris is acting downward with a force of 287 N, and the weight of the chair is acting downward with a force of 223 N.
The force Chris pulls on the rope is acting upward with a force of 265 N.
To calculate the net force, we need to determine whether the forces are acting in the same direction or opposite directions. In this case, the forces are acting in opposite directions.
Therefore, the net force can be found by subtracting the forces in the opposite direction:
Net force = (287 N + 223 N) - 265 N
Net force = 510 N - 265 N
Net force = 245 N
Now, we can use Newton's second law of motion, which states that the net force is equal to the mass of the system multiplied by the acceleration.
The mass of the system is the combined weight of Chris and the chair divided by the acceleration due to gravity. Therefore:
Mass of the system = (287 N + 223 N) / 9.8 m/s^2
Mass of the system = 510 N / 9.8 m/s^2
Mass of the system ≈ 52.04 kg
Now, we can rearrange Newton's second law to solve for acceleration:
Acceleration = Net force / Mass of the system
Acceleration = 245 N / 52.04 kg
Acceleration ≈ 4.71 m/s^2
Therefore, the magnitude of the acceleration of the system is approximately 4.71 m/s^2.
To find the magnitude of the force Chris exerts on the chair, we can use the concept of Newton's third law of motion, which states that every action has an equal and opposite reaction.
The force Chris exerts on the chair, which is pulling on the rope, is the same magnitude as the force the chair exerts on Chris in the opposite direction.
Therefore, the magnitude of the force Chris exerts on the chair is equal to 265 N.
So, the magnitude of the force Chris exerts on the chair is 265 N.
The rope it attached to the man’s hands, and the chair he is in. So,
there are two tensions pulling upward on the “child-chair” system.
m=(287+223)/g=510/g,
T=265 N
2Tmg ma
ma= ( 2T-mg)/m =
=g (2•265- 510)/510=
=9.8(530-510)/510=
=0.38 m/s²
N=mg+ma = 287 + 287•0.38/9.8 = 298.13 N