Exercise

1. Resultant Vector

A resultant vector is a single vector that has the same effect as the combined effect of all the vectors to be added.

                                                  OR

The sum of two or more vectors is a single vector, which has the same effect as the combined effect of all the vectors to be added. This single vector is called the resultant vector.

2. Torque

The turning effect of a force is called torque or moment of the force.

$\tau=F\times L$

Torque is a vector quantity and its direction can be found by using the right and rule.

Unit of torque

The unit of torque is Nm

3. Centre of Mass

Centre of the mass of a system is such a point where an applied force causes the system to move without rotation.

4. Centre of Gravity

A point where the whole weight of the body appears to act vertically downward is called the Centre of gravity of a body.

Difference between Torque and Couple:

Torque is a special kind of force that can rotate an object about an axis. While a force is described as a pull or push, it is better to think of torques as a twist

In a special case when applied force vectors add to zero, then the force is called a couple and their moment is called torque. Thus, the rotational force produces no moment between the two forces. The moment of a couple is called a torque.

When a driver turns a steering wheel, he exerts two equal but opposite forces on it. The two forces form a couple. The turning effect of a couple is the sum of the moment of the two forces. The moment of a couple is called Torque.

(i) Difference between stable and neutral equilibrium

States of equilibrium:

There are three states of equilibrium

• Stable equilibrium
• Unstable equilibrium
• Neutral equilibrium

1. Stable equilibrium:

When the Centre of gravity of a body lies below the point of the suspension or support, the body is said to be in stable equilibrium. For example, a book lying on the table is in stable equilibrium.

Explanation:

A book lying on a horizontal surface is an example of stable equilibrium. If the book is lifted from one edge and then allowed to fall, it will come back to its original position. Other examples of stable equilibrium are bodies lying on the floor such as a chair, table etc.

Reason for stability:

When the book is lifted, its centre of gravity is raised. The line of action of weight passes through the base of the book. The torque due to the weight of the book brings it back to the original position.

2. Unstable Equilibrium:

When the Centre of gravity of a body lies above the point of suspension or support, the body is said to be in unstable equilibrium.

Example

Pencil standing on its pint or a stick in a vertically standing position.

Explanation:

If a thin rod standing vertically is slightly disturbed from its position it will not come back to its original position. This type of equilibrium is called unstable equilibrium, other examples of unstable equilibrium are vertically standing cylinders and funnels.

Reason for instability:

When the rod is slightly disturbed its Centre of gravity is lowered. The line of action of its weight lies outside the base of the rod. The torque due to the weight of the rod toppled it down.

3. Neutral equilibrium:

When the Centre of gravity of a body lies at the point of suspension or support, the body is said to be in neutral equilibrium.

Explanation:

If a ball is pushed slightly to roll, neither will it come back to its original position nor it will roll forward rather it will remain at rest. This type of equilibrium is called a neutral equilibrium.

Reason for neutral equilibrium:

If a ball is rolled, its Centre of gravity is neither raised nor lowered. This means that its Centre of gravity is at the same height as before.

Difference between like and unlike forces:

Addition of vectors by head to the tail rule:

To add the vectors, draw the representative lines of these vectors in such a way that the head of the first vector coincides with the tail of the second vector. The line joining the tail of the first vector with the head of the second vector represents the resultant vector. The direction of the resultant vector is from the tail of the first vector towards the head of the second. This is called the head-to-tail rule.

Note: It should note that the head to the tail rule can be used to add any number of forces. The vector representing the resultant force gives the magnitude and direction of the resultant force.

Resolution of forces/vectors:

The process of splitting up vectors into their component forces is called the resolution of forces.

OR

Splitting up a force into two mutually perpendicular components is called the resolution of that force. Vector resolution is reverse from vector addition.

Perpendicular component/rectangular components:

Consider a force F represented by line OA making an angle with an x-axis. Draw a perpendicular AB on the x-axis from A. According to the head-to-tail rule, OA is the resultant vector represented by OB and BA.

Thus

OA=OB+BA………………. (1)

From figure

F= $F_{x}+ F_{y}$ ………………… (2)

Magnitude of horizontal component:

In right angled triangle OBA

$\cos \theta=\frac{base}{hypotenuse}=\frac{OB}{OA}$

$\frac{Fx}{F}= \cos\theta$

$Fx=F \cos\theta$ …………………….. (3)

Magnitude of vertical component (F_y):

$\sin\theta=\frac{perpendicular}{hypotenuse}=\frac{BA}{OA}$

$\frac{Fy}{F}= \sin\theta$ 

F_y= F Sin ……………… (4)

Equations (3) and (4) give the magnitude of horizontal and rectangular components.

Trigonometric Table

Mini Exercise

In a right-angled triangle length of the base is 4 cm and its perpendicular is 3 cm. Find:

(i) Length of hypotenuse  (ii) sin θ

(iii)     cos θ                     (iv)     tan θ

Solution:

(i) Length of hypotenuse:

Pythagoras theorem:

$\left(Hypotenuse\right)^{2}= \left(Base\right)^{2}+\left(Perpendicular\right)^{2}$

$\left(Hypotenuse\right)^{2}= \left(4\right)^{2}+\left(3\right)^{2}$

$\left(Hypotenuse\right)^{2}= 16+9$

$\left(Hypotenuse\right)^{2}= 25$ by taking square root on both sides

Hypotenuse = 5 cm

(ii) $\sin\theta$:

$\sin\theta= \frac{Perpendicular}{Hypotenuse}= \frac{3}{5}$

(iii) $\cos\theta$:

$\cos\theta= \frac{Base}{Hypotenuse}= \frac{4}{5}$

(iv) $\tan\theta$:

$\tan\theta= \frac{Perpendicular}{Base}= \frac{3}{4}$

A body is said to be in equilibrium if no net force acts on it. A body in equilibrium thus remains at rest or moves with uniform velocity.

Examples:

      A car moving with uniform velocity on a leveled road and an airplane flying in the air with uniform velocity is an example of bodies in equilibrium

Conditions of equilibrium:

In the above examples, we see that a body at rest or in uniform motion is in equilibrium if the resultant force acting on it is zero. For a body in equilibrium, it must satisfy certain conditions. There are two conditions of equilibrium.

The first condition of equilibrium:

                    A body is said to satisfy the first condition of equilibrium if the resultant of all the forces acting on it is zero.

Explanation:

Let n number of forces F₁, F₂, F₃,…………., F_n is acting on a body such that

                           F₁+F₂+F₃+……….+F_n  = 0

Or 

∑ F= 0……………. (1)

The symbol ∑ is a Greek letter called Sigma used for summation. Equation (1)

Is called the first condition of equilibrium.

The first condition of equilibrium can also be stated in terms of the x and y components.

F_{1x}+F_{2x}+F_{3x}.....+F_{nx} =0

And 

F_{1y}+F_{2y}+F_{3y}.....+F_{ny} =0

Or

∑ Fx = 0

And 

∑ Fy  = 0

Examples:

A book lying on the table or a picture hanging on the wall, is at rest and thus satisfies the first condition of equilibrium and is thus in equilibrium.

   A paratrooper coming down with terminal velocity (constant velocity) also satisfies the first condition for equilibrium and is thus in equilibrium.

A body satisfies the second condition for equilibrium when the resultant torque acts on its zero. Mathematically

∑ τ = 0

A car moving with uniform velocity on a leveled road and an airplane flying in the air with uniform velocity is an example of bodies in equilibrium

A paratrooper coming down with terminal velocity (constant velocity) also satisfies the first condition for equilibrium and is thus in equilibrium.

Rest implies stationary equilibrium implies a resultant force of zero. Therefore, a body in equilibrium could be moving. For example a skydiver at terminal velocity, where resistive forces are equal to the force of gravity. This means that a body can be in equilibrium and not at rest, but a body at rest must be in equilibrium. Otherwise, it would move so to answer the question: it is impossible.

No, with only a single force present, the body will accelerate infinitely in the direction of the force.

Because the force that is alone applied will have, some direction, and the object will try to move in this direction under its influence. However, if two opposite and equal forces take part it gives rise to a null vector force. The body can be in rotational equilibrium under the impact of a single force.

On the whole, the weight of the body acts as the center of gravity so, in the case of a racing car center of gravity must be close to the earth so that there are fewer chances of the overturning of the car.

If the car is high, it is easy to produce the torque in the car due to a large moment arm, and the car can take a somersault (forward roll)

States of equilibrium:

There are three states of equilibrium: stable equilibrium, unstable equilibrium, and neutral equilibrium.

1. Stable equilibrium:

A body is said to be in stable equilibrium if after a slight tilt it returns to its previous position.

Example:

Consider a book lying on the table. Tilt the book slightly about its one edge by lifting it from the opposite side as shown in the figure. It returns to its previous position when set free. Such a state of the body is called stable equilibrium.

Features of stable equilibrium:

When a body is in stable equilibrium, its center of gravity is at the lowest position. When it is tilted, its center of gravity rises. It returns to its stable state by lowering its center of gravity. A body remains in stable equilibrium as long as the center of gravity acts through the base of the body.

Explanation:

Consider a block shown in the figure. When the block is tilted, its center of gravity G rises. If the vertical line through G passes through its base in the tilted position as shown in figure (b), the block returns to its previous position. If the vertical line through G gets out of its base as shown in figure(c), the block does not return to its previous position.

2. Unstable equilibrium:

If a body does not return to its previous position when set free after the slightest tilt is said to be in unstable equilibrium.

Example:

Take a pencil and try to keep it in the vertical position on its tip as shown in the figure. Whenever you leave it, the pencil topples over about its tip and falls. This is called an unstable equilibrium. Thus, a body is unable to keep itself in a state of unstable equilibrium.

Features of unstable equilibrium:

The center of gravity of the body is at its highest position in a state of unstable equilibrium. As the body topples over about its base (tip), its center of gravity moves towards its lower position and does not return to its previous position.

DO YOU KNOW?

Vehicles are made heavy at the bottom. This lowers their center of gravity and helps to increase their stability.

3. Neutral equilibrium:

If a body remains in its new position when disturbed from its previous position, it is said to be in a state of neutral equilibrium.

Example:

Take a ball and place it on a horizontal surface as shown in the figure. Roll the ball over the surface and leave it after displacing the fit ffrom its previous position. It remains in its new position and does not return to its previous position. This is called a neutral equilibrium. There are various objects which have neutral equilibrium such as a ball, a sphere, a roller, a pencil lying horizontally, an egg lying horizontally on a flat surface, etc.

Features of neutral equilibrium:

In neutral equilibrium, all the new states in which a body is moved are the stable states and the body remains in its new state. In neutral equilibrium, the center of gravity of the body remains at the same height irrespective of its new position.