A painter of mass 80.0 kg stands on a platform of mass 75.0 kg and pulls on two ropes which hang over pulleys, as shown. He pulls each rope with a force of 400.0 N. Assuming a coordinate system where +y is up, calculate the y-component of the acceleration ay. ay can either be positive or negative depending on the direction of the acceleration.
To calculate the y-component of the acceleration (ay), we need to analyze the forces acting on the system.
Given:
- Mass of the painter (m1) = 80.0 kg
- Mass of the platform (m2) = 75.0 kg
- Force applied by the painter on each rope (F) = 400.0 N
First, let's find the net force acting on the system in the y-direction:
1. The weight of the painter (m1 * g) pulls downwards with a force of (80.0 kg) * (9.8 m/s^2) = 784.0 N.
2. The weight of the platform (m2 * g) pulls downwards with a force of (75.0 kg) * (9.8 m/s^2) = 735.0 N.
Since these forces act in the opposite direction of the positive y-axis, they need to be written as negative values in our calculation.
The net force in the y-direction is:
F_net = F - (m1 * g) - (m2 * g)
= 400.0 N - 784.0 N - 735.0 N
= -1119.0 N
Now, we can apply Newton's second law in the y-direction, which states that the net force is equal to the mass multiplied by the acceleration:
F_net = (m1 + m2) * ay
Rearranging the equation, we can solve for ay:
ay = F_net / (m1 + m2)
= -1119.0 N / (80.0 kg + 75.0 kg)
= -1119.0 N / 155.0 kg
≈ -7.22 m/s²
Hence, the y-component of the acceleration (ay) is approximately -7.22 m/s². Note that it is negative, indicating that the system is accelerating downwards.