Centre of Gravity, Torque and Equilibrium: Exam Notes
Physics notes on centre of gravity, torque and equilibrium for exams: moment of a force, couples, levers, conditions of equilibrium and types of stability.
By GK24 Editorial Team· Published · 6 min read
A force can do two things to a rigid body: move it from place to place, or turn it about an axis. The turning effect is torque, the point at which the whole weight of a body may be taken to act is its centre of gravity, and a body in which both effects cancel is in equilibrium. These three ideas run through every question on seesaws, spanners, levers, loaded buses and leaning towers, and the facts examiners want are few and fixed: the formula for torque, its unit, where the centre of gravity of standard bodies lies, and the two conditions of equilibrium.
Centre of mass and centre of gravity
The centre of mass of a body is the point at which its whole mass may be taken to be concentrated for the purpose of motion. The centre of gravity is the point through which the resultant weight of the body acts. In a uniform gravitational field, which is the case for all ordinary bodies on the earth, the two points coincide, and papers treat them as the same; they separate only for bodies so large that gravity is measurably stronger at the lower end. Two facts about the centre of gravity are asked again and again. First, it need not lie within the material of the body: it lies at the centre of the hole in a ring, at the centre of the cavity of a hollow sphere and outside the metal of an L shaped lamina or a horseshoe. Second, a suspended body hangs so that its centre of gravity lies vertically below the point of suspension, which is the basis of the plumb line method of finding it for an irregular lamina.
Centre of gravity of common uniform bodies
| Body | Position of the centre of gravity |
|---|---|
| Uniform straight rod | Its mid point |
| Rectangular or square lamina | The point of intersection of the diagonals |
| Circular ring or disc | The centre of the circle |
| Triangular lamina | The centroid, where the medians meet, at one third of the height from the base |
| Solid sphere or hollow sphere | The centre of the sphere |
| Solid right circular cone of height h | On the axis, at h divided by 4 above the base |
| Solid hemisphere of radius R | On the radius of symmetry, at 3R divided by 8 from the centre of the flat face |
| Hollow hemisphere of radius R | At R divided by 2 from the centre of the flat face |
| Uniform cylinder | The mid point of its axis |
Torque, the moment of a force
Torque measures the turning effect of a force about an axis. If a force F acts at a point whose position vector from the axis is r, and the angle between them is theta, then the magnitude of the torque is r into F into sine theta, which can be read as force multiplied by the perpendicular distance of its line of action from the axis. Torque is a vector, directed along the axis of rotation and given by the right hand rule. Its SI unit is the newton metre, and its dimensional formula is M L squared T to the power minus two, the same as that of work and energy, although torque is a different physical quantity. Three consequences are worth memorising. A force whose line of action passes through the axis produces no torque, because theta is zero. Torque is greatest when the force is applied at right angles to the arm. And for the same force, a longer arm gives a larger torque, which is why a long spanner loosens a tight nut, why a door handle is fixed far from the hinges, and why the steering wheel of a heavy vehicle is large. Work done by a constant torque is torque into angular displacement, and the power delivered is torque into angular velocity.
Couple and the principle of moments
Two equal and opposite parallel forces whose lines of action do not coincide form a couple. A couple produces no translation, only rotation, and the moment of a couple is one force into the perpendicular distance between the two lines of action, as in turning a tap, a screwdriver or a steering wheel. The principle of moments states that a body free to rotate about a fixed axis is in rotational equilibrium when the sum of the clockwise moments about that axis equals the sum of the anticlockwise moments. This is the working rule of the beam balance, the seesaw and every lever. For a lever, the mechanical advantage is the ratio of the load to the effort, which equals the ratio of the effort arm to the load arm, so a long effort arm multiplies force.
| Class of lever | Arrangement | Examples |
|---|---|---|
| First class | Fulcrum between effort and load | Seesaw, beam balance, crowbar, scissors, pliers |
| Second class | Load between fulcrum and effort | Wheelbarrow, nutcracker, bottle opener, lemon squeezer |
| Third class | Effort between fulcrum and load | Forceps, sugar tongs, fishing rod, the human forearm |
The conditions of equilibrium
A rigid body is in complete mechanical equilibrium when two conditions hold together: the vector sum of all the forces on it is zero, which gives translational equilibrium, and the vector sum of all the torques about any axis is zero, which gives rotational equilibrium. Either alone is not enough, and the standard example is a couple: the forces cancel, so the body does not move away, yet it spins because the torques do not cancel.
Stable, unstable and neutral equilibrium
- Stable equilibrium: on a slight displacement the centre of gravity rises and the body returns to its old position. Examples are a cone resting on its base, a book lying flat and a ship with ballast low down.
- Unstable equilibrium: on a slight displacement the centre of gravity falls and the body moves further away. Examples are a cone balanced on its apex, a pencil standing on its point and an egg on its narrow end.
- Neutral equilibrium: the centre of gravity stays at the same height and the body rests in the new position. Examples are a sphere or a cylinder lying on a horizontal plane and a cone resting on its side.
Stability in everyday life
A body is more stable when its centre of gravity is lower and the base on which it stands is wider, and it topples when the vertical line through its centre of gravity falls outside the base. This single rule explains a long list of examination examples: racing cars are built low and wide; a double decker bus carries more passengers on the lower deck; a passenger standing in a moving bus spreads the feet to widen the base; a person carrying a load on the back leans forward to keep the combined centre of gravity over the feet; and the leaning tower of Pisa stands because the vertical line through its centre of gravity still falls within its base. In rotation, the counterpart of mass is the moment of inertia, the sum of mass into the square of the distance from the axis, so torque equals moment of inertia into angular acceleration and angular momentum is moment of inertia into angular velocity. When no external torque acts, angular momentum is conserved, which is why a skater who pulls the arms inwards reduces the moment of inertia and spins faster, and why a diver tucks the body to turn quickly in the air.
Exam Point of View
Expect one or two questions in every general science paper. The commonest are the SI unit of torque, the formula with the sine of the angle, the position of the centre of gravity of a ring, a cone or a triangular lamina, and the classification of a given tool as a first, second or third class lever, where the wheelbarrow and the forceps are the usual choices. Conceptual questions ask the two conditions of equilibrium together, or test the couple as the case where forces cancel but rotation continues. The stability rule is asked through daily life examples, such as why racing cars are low and wide and why a standing passenger spreads the feet. Numerical questions are simple: multiply the force by the perpendicular arm, or balance the moments on the two sides of a pivot.
Important Facts
| Torque | r F sine theta, that is force into perpendicular distance from the axis |
|---|---|
| SI unit of torque | Newton metre |
| Dimensional formula | M L squared T to the power minus two |
| Nature of torque | Vector quantity, directed along the axis by the right hand rule |
| Zero torque | When the line of action of the force passes through the axis |
| Moment of a couple | One force into the perpendicular distance between the two lines of action |
| Principle of moments | Sum of clockwise moments equals sum of anticlockwise moments |
| Mechanical advantage of a lever | Load divided by effort, equal to effort arm divided by load arm |
| First class lever | Fulcrum in the middle: seesaw, beam balance, scissors |
| Second class lever | Load in the middle: wheelbarrow, nutcracker, bottle opener |
| Third class lever | Effort in the middle: forceps, sugar tongs, human forearm |
| Centre of gravity of a ring | At the centre of the ring, outside its material |
| Centre of gravity of a solid cone | On the axis at h divided by 4 from the base |
| Centre of gravity of a triangular lamina | At the centroid, one third of the height from the base |
| Conditions of equilibrium | Net force zero and net torque zero |
| Condition for stability | Low centre of gravity and wide base; vertical through the centre of gravity inside the base |
Practice MCQs on this topic
What is the SI unit of torque?
- A.Newton
- B.Newton metre
- C.Newton per metre
- D.Joule per second
Show answer
Torque is the product of a force and the perpendicular distance of its line of action from the axis of rotation, so its unit is the unit of force multiplied by the unit of length, the newton metre, and its dimensional formula is M L squared T to the power…
Read the full explanation →Three forces acting on a rigid body are represented in magnitude, direction and line of action by the three sides of a triangle taken in order. The forces are equivalent to a couple whose moment is equal to
- A.Thrice the area of the triangle
- B.Twice the area of the triangle
- C.The area of the triangle
- D.Half the area of the triangle
Show answer
When three forces are represented by the sides of a triangle taken in order, their vector sum is zero, because the three sides close the triangle, so the body cannot be pushed away; but the lines of action are different, so the system reduces to a couple.
Read the full explanation →The centre of gravity of a uniform circular ring lies
- A.On the ring itself, at any point
- B.At the centre of the ring, outside its material
- C.At one third of the radius from the centre
- D.Nowhere, because a ring has no centre of gravity
Show answer
The centre of gravity is a geometrical point through which the resultant weight of a body acts, and by symmetry that point for a uniform ring is its geometrical centre, which lies in the empty space inside the ring.
Read the full explanation →The torque produced by a force about an axis is maximum when the angle between the force and the arm is
- A.0 degree
- B.45 degrees
- C.90 degrees
- D.180 degrees
Show answer
The magnitude of torque is r into F into the sine of the angle between the arm and the force, and the sine of an angle is largest, equal to one, at 90 degrees, so the whole of the force is then used in turning the body.
Read the full explanation →A long spanner loosens a tight nut more easily than a short one because it
- A.Reduces the force needed by increasing the moment arm
- B.Reduces the friction at the nut
- C.Increases the mass acting on the nut
- D.Changes the direction of the weight of the nut
Show answer
Torque is the product of the force and the perpendicular distance of its line of action from the axis, so the same turning effect can be produced by a smaller force if that distance is made larger; a long handle therefore does the same work with less effort.
Read the full explanation →A rigid body is said to be in complete mechanical equilibrium when
- A.The net force on it is zero
- B.The net torque on it is zero
- C.Both the net force and the net torque on it are zero
- D.Its centre of gravity lies at its geometrical centre
Show answer
A zero net force gives translational equilibrium, so the body as a whole does not accelerate, and a zero net torque gives rotational equilibrium, so it does not begin to spin; complete equilibrium requires the two together.
Read the full explanation →When a body in stable equilibrium is slightly displaced, its centre of gravity
- A.Rises, and the body returns to its original position
- B.Falls, and the body moves further away
- C.Stays at the same height
- D.Shifts outside the body
Show answer
In stable equilibrium the original position is the one of least potential energy, so a small displacement lifts the centre of gravity and the weight of the body brings it back, as with a cone resting on its base or a book lying flat on a table.
Read the full explanation →A wheelbarrow is an example of which class of lever?
- A.First class
- B.Second class
- C.Third class
- D.It is not a lever
Show answer
In a wheelbarrow the wheel is the fulcrum, the load in the tray lies between the wheel and the hands, and the effort is applied at the handles, so the load sits in the middle, which defines a second class lever; a nutcracker, a bottle opener and a lemon squeezer…
Read the full explanation →The centre of gravity of a uniform solid right circular cone of height h lies on its axis at a distance of
- A.h divided by 2 from the base
- B.h divided by 3 from the base
- C.h divided by 4 from the base
- D.3h divided by 4 from the base
Show answer
A solid cone has more material near its wide base than near its apex, so its centre of gravity lies on the axis of symmetry well below the middle, at one quarter of the height measured from the base, which is three quarters of the height measured down from the…
Read the full explanation →A spinning skater pulls both arms close to the body and spins faster. This happens because
- A.The angular momentum of the skater increases
- B.The moment of inertia decreases, so the angular velocity increases
- C.The torque applied by the skater increases
- D.The mass of the skater decreases
Show answer
Moment of inertia is the sum of each bit of mass multiplied by the square of its distance from the axis, so drawing the arms in reduces it.
Read the full explanation →The following statements (S1 and S2) pertain to principle of moments stated as Varignon theorem: S1: Theorem states that the Square of the moment of a Force is equal to the Sum of the Square moment of its components. S2: For applying Varignon Theorem, the force system may be either coplanar or spatial, but it must be concurrent. Identify the statements as True or False and choose the best answer.
- A.Both S1 and S2 are False
- B.Both S1 and S2 are True
- C.S1 is True and S 2 is False
- D.S1 is False and S 2 is True
Show answer
Varignon theorem states that the moment of a force about any point equals the algebraic sum of the moments of its components about the same point; it speaks of the moments themselves, not of their squares, so statement S1 is false as written.
Read the full explanation →A force of 20 newton is applied at right angles to a rod at a point 0.5 metre from the axis of rotation. The torque about the axis is
- A.4 newton metre
- B.10 newton metre
- C.40 newton metre
- D.20.5 newton metre
Show answer
Torque is r into F into the sine of the angle between them, and since the force here is perpendicular to the rod the sine is one, so the torque is simply 0.5 multiplied by 20, that is 10 newton metre.
Read the full explanation →Frequently Asked Questions
What is the difference between centre of mass and centre of gravity?
The centre of mass is the point where the whole mass of a body may be taken to be concentrated, while the centre of gravity is the point through which the resultant weight acts. In a uniform gravitational field, as on the surface of the earth for ordinary bodies, the two coincide; they differ only for bodies so large that gravity varies over their height.
Why is it easier to loosen a nut with a longer spanner?
Torque is force into the perpendicular distance from the axis, so for the same push of the hand a longer spanner gives a longer arm and therefore a larger turning effect. The same reasoning explains why door handles are fixed far from the hinges and why heavy vehicles have large steering wheels.
Can the centre of gravity of a body lie outside the body?
Yes. The centre of gravity is a geometrical point, not a particle of matter, so it can lie in empty space: it is at the centre of the hole of a ring or a bangle, at the centre of the cavity of a hollow sphere, and outside the material in an L shaped lamina, a horseshoe or a boomerang.
Why does a body in a couple not move away even though forces act on it?
In a couple the two forces are equal, opposite and parallel, so their vector sum is zero and there is no translational motion. Their lines of action are different, however, so their moments add instead of cancelling, and the body rotates. This is why both conditions are needed for complete equilibrium.
Why are racing cars built low and wide?
Stability increases when the centre of gravity is lower and the base wider, because the vertical line through the centre of gravity then stays inside the base even when the vehicle tilts on a turn. A low, wide racing car is therefore far less likely to topple than a tall, narrow one at the same speed.
