In Figure 5-66, a force F of magnitude 13 N is applied to a FedEx box of mass m2 = 1.0 kg. The force is directed up a plane tilted by θ = 37°. The box is connected by a cord to a UPS box of mass 3.0 kg on the floor. The floor, plane, and pulley are frictionless, and the masses of the pulley and cord are negligible. What is the tension in the cord?
N
what pulleys?
To find the tension in the cord, we need to analyze the forces acting on both boxes.
Step 1: Identify the forces acting on the FedEx box:
a) Weight (mg): The weight of the FedEx box is given by the equation mg, where m is the mass of the box and g is the acceleration due to gravity.
b) Tension (T): The tension in the cord acts in the opposite direction to the weight of the FedEx box.
c) Normal force (N): The normal force acts perpendicular to the inclined plane. It counteracts the component of weight acting perpendicular to the inclined plane.
d) Friction force (f): Since the inclined plane and pulley are frictionless, there is no friction force.
Step 2: Apply Newton's Second Law:
For the FedEx box:
The force equation can be written as: ΣF = ma, where ΣF is the sum of the forces acting on the box, m is the mass, and a is the acceleration.
In the vertical direction:
ΣFy = T - mgcosθ = may (equation 1)
In the horizontal direction:
ΣFx = mgsinθ = max (equation 2)
Step 3: Solve the equations simultaneously:
From equation 2, we can isolate a:
a = gsinθ
Substitute this value into equation 1:
T - mgcosθ = m(gsinθ)
Simplify the equation:
T = m(gcosθ + gsinθ)
T = mg(cosθ + sinθ)
Step 4: Calculate the tension in the cord:
Given the mass of the FedEx box, m2 = 1.0 kg, and the angle of inclination, θ = 37°, substitute these values into the equation:
T = (1.0 kg)(9.8 m/s^2)(cos37° + sin37°)
Using trigonometric identities, we can calculate the values for cos37° and sin37°:
cos37° ≈ 0.7986
sin37° ≈ 0.6018
Substitute these values into the equation:
T ≈ (1.0 kg)(9.8 m/s^2)(0.7986 + 0.6018)
Simplify the equation:
T ≈ (1.0 kg)(9.8 m/s^2)(1.4004)
Calculate the tension:
T ≈ 13.72 N
Therefore, the tension in the cord is approximately 13.72 N.
To find the tension in the cord, you can use Newton's second law and the concept of equilibrium. Here's how you can approach the problem:
1. Draw a free-body diagram for each object in the system. In this case, we have the FedEx box and the UPS box.
2. Identify the forces acting on each object. For the FedEx box, we have the force applied (F), the gravitational force (mg), and the tension in the cord (T). For the UPS box, we only have the gravitational force (m * g).
3. Write down the equations of motion for each object along the x-axis and y-axis. Since the system is in equilibrium, the sum of forces in both directions should be zero.
For the FedEx box (in the direction perpendicular to the plane):
- Sum of forces in the y-direction: T - mg * cos(θ) = 0
- Solving for T: T = mg * cos(θ)
For the FedEx box (in the direction parallel to the plane):
- Sum of forces in the x-direction: -F + T * sin(θ) = 0
- Solving for T: T = F / sin(θ)
4. Now, substitute the given values into the equations. In this case, we know F = 13 N, m = 1.0 kg, θ = 37°, and g = 9.8 m/s^2.
Using the equation T = mg * cos(θ), we can find T:
T = (1.0 kg) * (9.8 m/s^2) * cos(37°)
Using the equation T = F / sin(θ), we can find T:
T = (13 N) / sin(37°)
5. Calculate the tension in the cord using either equation. The final answer should be the same regardless of which equation you use.
By substituting the values into the equations and solving, you will get the value of tension (T) in Newtons.
Note: Make sure to use the appropriate units when plugging in the values and calculating.