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
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
2T=mg +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
To solve this problem, we need to understand the forces acting on the system.
Let's break down the forces involved:
1. Chris's weight (Wc): This is the force pulling Chris towards the ground, given as 287 N.
2. The chair's weight (Wchair): This is the force pulling the chair towards the ground, given as 223 N.
3. The tension in the rope (Tension): This is the force applied by Chris to pull the rope, measured as 265 N.
Now, let's calculate the acceleration of the system (a):
The net force acting on the system is the difference between the tension and the combined weight of Chris and the chair:
Net force (Fnet) = Tension - (Wc + Wchair)
Plugging the values:
Fnet = 265 N - (287 N + 223 N)
= 265 N - 510 N
= -245 N (negative because the net force is upwards)
Since we have the net force, we can use Newton's second law of motion to find the acceleration (a):
Fnet = m * a
Where m is the total mass of the system. We can find the total mass by adding the masses of Chris and the chair:
m = (Wc + Wchair) / g
Plugging in the values:
m = (287 N + 223 N) / 9.8 m/s^2
= 510 N / 9.8 m/s^2
≈ 52.04 kg
Now, we can find the acceleration (a):
Fnet = m * a
-245 N = (52.04 kg) * a
a ≈ -245 N / 52.04 kg
a ≈ -4.71 m/s^2
Since we are asked for the magnitude of the acceleration, we take the absolute value:
Magnitude of acceleration = | a | ≈ 4.71 m/s^2 (answer to part a)
Now, let's calculate the magnitude of the force Chris exerts on the chair (Fchris):
Fchris = Tension - Wc
Plugging the values:
Fchris = 265 N - 287 N
= -22 N (negative because it's in the opposite direction to tension)
Again, taking the magnitude:
Magnitude of force Chris exerts on the chair = | Fchris | ≈ 22 N (answer to part b)