Solve the incline plane problem presented in class on Monday and assigned as part of HW #18 using the coordinate system shown below. In this problem, a 20 kg box is pulled up a 30◦ incline with a string. The box is moving at constant velocity. Find the normal force and the tension force. [Hint: With this choice of coordinate system, the force due to gravity will not have the x-component, etc.]
To solve the incline plane problem, we need to analyze the forces acting on the box. In this case, there are three main forces: gravity (mg), the normal force (N), and the tension force in the string (T).
First, let's define our coordinate system. We will use a coordinate system with the x-axis parallel to the incline and the y-axis perpendicular to the incline. This choice of coordinate system is suggested in the problem statement.
Now, let's break down the forces.
1. Gravity (mg):
Gravity always acts vertically downward. So, in our chosen coordinate system, the force due to gravity does not have an x-component. The force due to gravity only has a y-component.
Given that the mass of the box is 20 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the force due to gravity as follows:
Force due to gravity (Fg) = mass (m) * acceleration due to gravity (g)
Fg = 20 kg * 9.8 m/s^2 = 196 N
So, the force due to gravity has a y-component of -196 N (negative sign indicates downward direction).
2. Normal Force (N):
The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force acts perpendicular to the incline.
Since the box is moving at constant velocity, we know that it is not accelerating in the y-direction. Therefore, the net force in the y-direction must be zero.
Net force in the y-direction = normal force (N) - force due to gravity (Fg y-component) = 0
Rearranging the equation, we can find the normal force:
Normal force (N) = force due to gravity (Fg y-component)
N = 196 N
So, the normal force is 196 N and it acts perpendicular to the incline.
3. Tension Force (T):
The tension force is the force transmitted through a string, rope, cable, or any other type of flexible connector. In this case, the tension force is acting along the incline.
Since the box is moving at constant velocity, we know that the net force in the x-direction must also be zero.
Net force in the x-direction = tension force (T) - force due to gravity (Fg x-component) = 0
However, in our chosen coordinate system, the force due to gravity does not have an x-component. Therefore, the net force in the x-direction is just equal to the tension force.
Net force in the x-direction = T = 0
So, the tension force is zero.