Not drawing a free body diagram

Oct 13, 2025 by wbrum.org 117 views

Introduction

In this problem, we have two objects connected by an inextensible string, which are being pulled up an incline by a force F. The incline makes an angle of 30° with the horizontal. We are given the masses of the two objects as 2 kg and 0.8 kg, respectively. Our goal is to calculate the force F required to pull the objects up the incline.

Given Information

Mass of object 1 (m1) = 2 kg
Mass of object 2 (m2) = 0.8 kg
Angle of incline (θ) = 30°
Ignore the mass of the string and the effects of friction

Free Body Diagram

To solve this problem, we need to draw a free body diagram of the two objects. The free body diagram shows the forces acting on each object.

Object 1

Weight (W1) = m1 × g
Tension (T) = F (since the string is inextensible)
Normal force (N1) = 0 (since the object is on an incline)

Object 2

Weight (W2) = m2 × g
Tension (T) = F (since the string is inextensible)
Normal force (N2) = 0 (since the object is on an incline)

Resolving Forces

We need to resolve the forces acting on each object into their x and y components.

Object 1

Weight (W1) = m1 × g = 2 kg × 9.8 m/s² = 19.6 N
Tension (T) = F (since the string is inextensible)
Normal force (N1) = 0 (since the object is on an incline)
x-component of weight (W1x) = W1 × sin(θ) = 19.6 N × sin(30°) = 9.8 N
y-component of weight (W1y) = W1 × cos(θ) = 19.6 N × cos(30°) = 16.7 N

Object 2

Weight (W2) = m2 × g = 0.8 kg × 9.8 m/s² = 7.84 N
Tension (T) = F (since the string is inextensible)
Normal force (N2) = 0 (since the object is on an incline)
x-component of weight (W2x) = W2 × sin(θ) = 7.84 N × sin(30°) = 3.92 N
y-component of weight (W2y) = W2 × cos(θ) = 7.84 N × cos(30°) = 6.73 N

Equilibrium Equations

Since the objects are in equilibrium, the net force acting on each object must be zero.

Object 1

Net force in x-direction (Fx) = T × sin(θ) - W1x = 0
Net force in y-direction (Fy) = T × cos(θ) - W1y = 0

Object 2

Net force in x-direction (Fx) = T × sin(θ) - W2x = 0
Net force in y-direction (Fy) = T × cos(θ) - W2y = 0

Solving for T

We can solve for T by equating the x and y components of the forces.

Object 1

T × sin(θ) = W1x
T × cos(θ) = W1y

Object 2

T × sin(θ) = W2x
T × cos(θ) = W2y

Solving for F

Since the string is inextensible, the tension T is the same for both objects. We can solve for F by equating the x and y components of the forces.

F × sin(θ) = W1x + W2x
F × cos(θ) = W1y + W2y

Substituting Values

We can substitute the values of W1x, W1y, W2x, and W2y into the equations.

F × sin(30°) = 9.8 N + 3.92 N
F × cos(30°) = 16.7 N + 6.73 N

Solving for F

We can solve for F by dividing both sides of the equations by sin(30°) and cos(30°), respectively.

F = (9.8 N + 3.92 N) / sin(30°)
F = (16.7 N + 6.73 N) / cos(30°)

Calculating F

We can calculate F by plugging in the values.

F = (13.72 N) / 0.5
F = (23.43 N) / 0.866

Final Answer

F = 31.4 N

Q&A

Q: What is the purpose of the free body diagram in this problem? A: The free body diagram is a visual representation of the forces acting on each object. It helps us to identify the forces and their directions, which is essential for solving the problem.

Q: Why do we need to resolve the forces into their x and y components? A: We need to resolve the forces into their x and y components because the objects are on an incline. The forces acting on each object are not purely in the x or y direction, but rather at an angle to the horizontal. By resolving the forces into their x and y components, we can analyze the forces more easily.

Q: What is the significance of the angle of incline (θ) in this problem? A: The angle of incline (θ) is crucial in this problem because it affects the direction and magnitude of the forces acting on each object. The angle of incline determines the x and y components of the forces, which in turn affects the net force acting on each object.

Q: Why do we need to equate the x and y components of the forces? A: We need to equate the x and y components of the forces because the objects are in equilibrium. The net force acting on each object must be zero, which means that the x and y components of the forces must balance each other out.

Q: What is the relationship between the tension (T) and the force (F)? A: The tension (T) is equal to the force (F) because the string is inextensible. The tension in the string is the same for both objects, and it is equal to the force applied to the objects.

Q: How do we calculate the force (F) required to pull the objects up the incline? A: We calculate the force (F) by equating the x and y components of the forces and solving for F. We use the values of W1x, W1y, W2x, and W2y to substitute into the equations and solve for F.

Q: What is the final answer for the force (F) required to pull the objects up the incline? A: The final answer for the force (F) required to pull the objects up the incline is 31.4 N.

Common Mistakes

Not drawing a free body diagram: A free body diagram is essential for identifying the forces and their directions.
Not resolving the forces into their x and y components: Resolving the forces into their x and y components is crucial for analyzing the forces acting on each object.
Not equating the x and y components of the forces: Equating the x and y components of the forces is necessary for ensuring that the objects are in equilibrium.
Not using the correct values for W1x, W1y, W2x, and W2y: Using the correct values for W1x, W1y, W2x, and W2y is essential for calculating the force (F) required to pull the objects up the incline.

Conclusion

In conclusion, the problem of two objects being pulled up an incline by a force F is a classic physics problem that requires a thorough understanding of forces and equilibrium. By drawing a free body diagram, resolving the forces into their x and y components, and equating the x and y components of the forces, we can calculate the force (F) required to pull the objects up the incline. The final answer for the force (F) required to pull the objects up the incline is 31.4 N.