1.3 Forces in Equilibrium
An object is in equilibrium of forces when the forces acting on it produce a zero resultant force. Learn how to sketch triangles of forces and solve equilibrium problems using resolution of forces and scaled drawings.
Learning Objectives
- State the meaning of forces in equilibrium.
- Sketch a triangle of forces for three forces acting on an object in equilibrium.
- Demonstrate forces in equilibrium using a Vector Force Table Kit.
- Solve problems involving forces in equilibrium using the resolution of forces method and the scaled drawing method.
Triangle of Forces and Resolution of Forces for an Object on an Inclined Plane
See how three coplanar forces in equilibrium form a closed vector triangle and how forces are resolved parallel and perpendicular to an inclined plane.
Closed Triangle of Forces
Three non-parallel forces in equilibrium can be drawn tip-to-tail to form a closed triangle. The lengths of the sides represent the magnitudes of the forces.
Resolution on an Inclined Plane
Weight \(W\) is resolved into components parallel and perpendicular to the plane. Normal reaction \(R = W\cos\theta\) and friction \(F_R = W\sin\theta\).
Short Explanation
What is Forces in Equilibrium?
An object is in equilibrium of forces when the forces acting on it produce a zero resultant force. According to Newton's Second Law, \(F = ma\), when \(a = 0\), the resultant force is zero.
Stationary or Uniform Velocity
An object in equilibrium can be stationary or moving with uniform velocity. In both cases, acceleration is zero, so the resultant force is zero.
Triangle of Forces
For three non-parallel forces in equilibrium, the force vectors drawn in sequence tip-to-tail form a closed triangle of forces.
Resolution of Forces
Resolution of forces splits an inclined force into horizontal and vertical components. For equilibrium, \(\sum F_x = 0\) and \(\sum F_y = 0\).
Inclined Plane Components
For a plane inclined at angle \(\theta\) to the horizontal, the component of weight perpendicular to the plane is \(W\cos\theta\) and the component parallel to the plane is \(W\sin\theta\). Therefore, normal reaction \(R = W\cos\theta\) and frictional force \(F_R = W\sin\theta\).
Try to Answer First
Answer in your mind, then press “Check Answer”.
What is meant by forces in equilibrium?
Check Answer
If an object is acted on by three forces in equilibrium, what shape do the force vectors form when drawn in sequence head-to-tail?
Check Answer
A box of weight \(50\text{ N}\) is stationary on an inclined plane of \(40^\circ\). What is the magnitude of the frictional force acting on the box?
Check Answer
Common Mistakes
- Confusing the sine and cosine components on an inclined plane: taking \(W\cos\theta\) as parallel to the plane instead of perpendicular to the plane.
- Drawing arrows in a triangle of forces pointing against each other instead of following in a continuous tip-to-tail loop.
- Assuming that an object in equilibrium must always be stationary, forgetting that an object moving at a constant velocity is also in equilibrium.
- In scaled drawings, omitting the chosen scale or incorrectly converting lengths from cm back to forces in N.
Concept Misunderstandings
If an object is moving, forces cannot be in equilibrium.
If an object is moving at constant velocity in a straight line, its acceleration is zero. By \(F = ma\), the resultant force is zero, so the forces are in equilibrium.
The normal reaction \(R\) on an inclined plane is always equal to the weight \(W\).
The normal reaction is perpendicular to the inclined surface, so it balances only the perpendicular component of weight, \(R = W\cos\theta\), which is less than \(W\).
Summary
- Equilibrium of forces occurs when resultant force \(= 0\text{ N}\) (\(a = 0\text{ m s}^{-2}\)).
- Three coplanar forces in equilibrium can be represented by a closed triangle of forces drawn head-to-tail.
- For an inclined plane with angle \(\theta\) to the horizontal: component parallel to plane \(W\sin\theta = F_R\), component perpendicular to plane \(W\cos\theta = R\).
- Equilibrium problems can be solved using component resolution (\(\sum F_x = 0\), \(\sum F_y = 0\)), sine/cosine rules, or a scaled vector triangle.
Short Activity
Objective: Check your understanding of forces in equilibrium, triangle of forces and resolution of forces.
A. Objective Quiz
1 An object is said to be in equilibrium of forces when:
2 When three forces of equal magnitude act on an object such that the object is in equilibrium, what is the angle between each adjacent force?
B. Fill in the Blanks
3 For a stationary box of weight \(W\) resting on a plane inclined at angle \(\theta\) to the horizontal, the normal reaction \(R\) is balanced by the component of weight equal to .
4 When three forces acting on an object are in equilibrium, the vectors drawn in sequence tip-to-tail will form a closed .
C. Matching / Drag and Drop
Match each physical force on an inclined plane or hanging system to its correct description or expression. If using a phone, tap the answer first, then tap the matching box.
