Equilibrium of a Rigid Body Class 11 Notes

Equilibrium of a rigid body explains how it maintains its state by balancing forces to prevent motion and torques to prevent rotation. 

A single particle needs only force balance. But a rigid body demands both translational and rotational stability. This dual requirement for the rigid body to be in equilibrium is important to understand for two reasons. 

1. We can better explain systems like seesaws or suspended rods. 
2. We can better define how the placement of forces matters, just as their magnitude.

For school exams and even for approaching JEE Main prep, remember that 

• Translational equilibrium prevents linear acceleration.
• Rotational equilibrium prevents angular acceleration.
• Equilibrium can be static (at rest) or dynamic (constant velocity)

Section 6.8 of the Systems of Particles and Rotational Motion Chapter explains that 

“A rigid body is said to be in mechanical equilibrium, if both its linear momentum and angular momentum are not changing with time, or equivalently, the body has neither linear acceleration nor angular acceleration.”

The explanation for these conditions are 

• Translational Equilibrium: The net external force is zero: $\sum {\vec{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}=0$

This ensures no acceleration in any direction.

• Rotational Equilibrium: The net external torque about any point is zero: $\sum {\vec{\tau }}_{\mathrm{e}\mathrm{x}\mathrm{t}}=0$

This prevents rotation.

So, if you recall, torque is defined as

$\vec{\tau }=\vec{r}\times \vec{F}$

Here $\vec{r}$ is the position vector from the pivot to the force's point of application.

This is because zero torque about one point indicates zero torque about all points.

Equilibrium stability depends on how a body responds to small displacements. This is determined by the centre of mass (COM) behaviour. 

The types are

• Stable Equilibrium of Rigid Body: A slight displacement raises the COM, and the body returns to its original position. Take the example of a cone on its base. 
• Unstable Equilibrium of Rigid Body: A slight displacement lowers the COM, and the body moves further away. Consider a pencil balanced on its tip.
• Neutral Equilibrium of Rigid Body: A displacement neither raises nor lowers the COM, and the body stays in the new position. For instance, a cylinder on its side.

If you are preparing for exams, use these steps to solve equilibrium in a rigid body. 

• Draw a free-body diagram (FBD).
• Don't forget to mark all forces and their points of application. 
• Apply $\sum \vec{F}=0$ in components ( $\sum F_x=0,\sum F_y=0$ ).
• Select a pivot point so that you know where unknown forces usually act.
• Apply $\sum \vec{\tau }=0$ as the rigid body is in equilibrium. This helps when you use a consistent sign convention for torques.
• Solve the resulting equations.

Try yourself on the example problem below. 

Example Problem: A uniform rod of mass 3 kg and length 2 m is hinged at one end and supported by a string at the other, making ${60}^{\circ }$ with the horizontal. Find the tension.

 

Solution: Free-body diagram: Weight $mg=3\times 9.8=29.4\mathrm{N}$  at the centre, hinge force, and tension $T$ . Translational equilibrium: $\sum F_y=0:T\mathrm{s}\mathrm{i}\mathrm{n}{60}^{\circ }+F_{\text{hinge},y}-29.4=0$

Rotational equilibrium about the hinge: $\sum \tau =0:T\mathrm{c}\mathrm{o}\mathrm{s}{60}^{\circ }\times 2-29.4\times 1=0\Rightarrow T\times 0.5\times 2=29.4\Rightarrow T=29.4\mathrm{N}$

Find well-explained notes on all Physics chapters for Class 11. 

Units and Measurements Class 11 Notes	Mechanical Properties of Solids Class 11 Notes
Motion in a Straight Line Class 11 Notes	Mechanical Properties of Fluids Class 11 Notes
NCERT Class 11 Notes for Motion in a Plane	Thermal Properties of Matter Class 11 Notes
Laws of Motion Class 11 Notes	Thermodynamics Class 11 Notes
Work, Energy, and Power Class 11 Notes	Kinetic Theory of Gas Class 11 Notes
System of Particles and Rotational Motion Class 11 Notes	Oscillations Class 11 Notes
Gravitation Class 11 Notes	Waves Class 11 Notes

Get an overview of what you have to cover in Physics in Class 11, along with Maths and Chemistry. 

NCERT Class 11 Notes for PCM

NCERT Class 11 Physics Notes

 

Find NCERT Solutions for Physics Class 11. For each Physics chapter, you can practice these along.  

Units and Measurements Class 11 NCERT Solutions	Mechanical Properties of Solids Class 11 NCERT Solutions
Motion in a Straight Line Class 11 NCERT Solutions	Mechanical Properties of Fluids Class 11 NCERT Solutions
Motion in a Plane Class 11 NCERT Solutions	Thermal Properties of Matter Class 11 NCERT Solutions
Laws of Motion Class 11 NCERT Solutions	Thermodynamics Class 11 NCERT Solutions
Work, Energy, and Power Class 11 NCERT Solutions	Kinetic Theory Class 11 NCERT Solutions
System of Particles and Rotational Motion Class 11 NCERT Solutions	Oscillations Class 11 NCERT Solutions
Gravitation Class 11 NCERT Solutions	Waves Class 11 NCERT Solutions