Notes

No, we cannot loosen the nut of the axel of a bike. Normally we use a spanner because a spanner increases the turning effect of the force which easily loosened the nut of the axle of a bike.

He is trying to balance himself on a wooden plank which is placed over a cylindrical pipe. Due to opening the arms, he is doing its center of mass as low as possible to make him stable.

Women and children keep themselves upright when carrying pitchers on their heads. The pitcher has a heavy semi-spherical base. When it is tilted, its center of mass rises. It returns to its upright position at which its center of mass is at its lowest. That is why women and children in the villages often carry pitchers with water on their heads.

To balance something, all you need to do is make sure that the center of gravity of the object is either directly above or directly below the pivot point. An example would be balancing the stick on the end of a finger with the stick pointing vertically up. If you do this you will find that the stick wants to fall over, and you need to keep moving your finger around to keep this from happening.

Parallel Forces:

          In a plane, if several forces act on a body such that their points of action are different but lines of action are parallel to each other, then these forces are called parallel forces.

Difference between like and unlike forces:

Like forces acting in the same direction increases the resultant force which moves the bus easily.

An apple is suspended by a string. The string is stretched due to the weight of the apple. The forces acting on it are; the weight of the apple acting vertically downwards and the tension in the string pulling it vertically upwards. The two forces are parallel but opposite to each other. These forces are called parallel forces.

In the figure, Forces F₁ and F₂ are also unlike parallel Forces, because they are parallel and opposite to each Other. But F₁ and F₂ are not acting along the same line and Hence, they are capable to rotate the body.

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

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.

Trigonometry:

Trigonometry is a branch of mathematics that deals with the properties of a right-angled triangle.

Trigonometric ratios:

Consider a right-angled triangle $\triangle ABC$ having $\theta$ at A.

$\sin \theta$ = Perpendicular/ Hypotenuse $\frac{BC}{AB}$

$\cos \theta$ = Base/Hypotenuse $\frac{AC}{AB}$

$\tan \theta$ = Perpendicular/Base = $\frac{BC}{AC}$

Note:

To remember trigonometric ratios, we use the following sentence:

            “Some people have – Curly brown hair – Through proper brushing”

Pythagoras theorem:

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

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) $\tan\theta$:

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

Determination of a Force or a vector from its Perpendicular Components:

Consider F_X and F_y as the perpendicular components of a force F. These perpendicular components F_X and F_y

 are represented by lines OP and PR respectively.

According to head to the tail rule:

 OR = OP + PR

Thus, OR will completely represent the force F whose x and y-components 

are F_X and F_y respectively. That is

F = F_X + F_y

The magnitude of resultant force/Magnitude of resultant vector:

The magnitude of the force F can be determined using the right-angled triangle OPR

As

$\left(OR\right)^{2}=\left(OP\right)^{2}+\left(PR\right)^{2}$

$F^{2}=Fx^{2}+Fy^{2}$

Hence

F= $\surd Fx^{2}+Fy ^{2}$ (i)

Direction of the resultant force/Direction of the resultant vector:

The direction of the force F with x-axis is given by

$\tan\theta=\frac{PR}{OP}=\frac{Fy}{Fx}$

$\theta=\tan^{-1}\frac{Fy}{Fx}$

We open or close a door by pushing or pulling it. Here push or pull to turn the door about its hinge or axis of rotation. The door is opened or closed due to the turning effect of the force acting on it.

Rigid Body:

            A body is composed of a large number of small particles. If the distances between all pairs of particles of the body do not change by applying a force then it is called a rigid body. In other words, a rigid body is not deformed by force or forces acting on it.

Axis of rotation:

            Consider a rigid body rotating about a line. The particles of the body move in circles with their centers all lying on this line. This line is called the axis of rotation of the body.

Turning a pencil in a sharpener, turning the stopcock of a water tap, turning the doorknob, and so on are some of the examples where a force produces a turning effect.

QUICK QUIZ

1: Name some more objects that work by the turning effects of forces:

(i)  Torque is produced when a force is applied to the paddle of a  bicycle. Because by applying force its wheels experience the  rotational effect (torque)

(ii) Torque is produced when a force is applied to the door to open.

Torque (moment of a force):

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

$Torque \: \tau= F\times L$

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

Unit of torque:

The unit of torque is Nm.

Torque depends upon two factors

The torque or moment of a force depends upon the force F and the moment arm L of the force.

1. Magnitude of the force(F)

Greater is a force, greater is the moment of the force.

Τ $\propto$ F   ………………………………………. (i)

2. Moment arm

Similarly, the longer is the moment arm, the greater is the moment of the force.

Τ $\propto$ L    ………………………………………. (ii)

We can open or close a door more easily by applying a force at the outer edge of a door rather than near the hinge.

            The moment produced by a force using a greater moment arm is greater than the torque produced by the same force by using a shorter moment arm.

Therefore, the handle of a door is fixed near the outer edge a door. (Τ ∝ L)

A spanner with having long arm helps to loosen or tighten a nut or a bolt with greater ease than the one with having short arm. It is because the turning effect (torque) of the force increases. (Τ $\propto$ L)

Line of action of a force:

The line along which a force acts is called the line of action of the force. In the figure, line BC is the lie of action of force F.

Moment arm:

The perpendicular distance between the axis of rotation and the line of action of the force is called the moment arm of the force. It is represented by the distance L.

SI unit of torque is newton-meter (Nm).

Newton-meter (Nm):

A torque of 1 N m is caused by a force of 1 N acting perpendicular to the moment arm 1 m long.

Mini Exercise

A force of 150 N can loosen a nut when applied at the end of a spanner 10 cm long.

Solution: 

F = 150 N

$L = 10 cm=\frac{10}{100}=0.1m$

Torque Τ = F × L

= 150 \: N\times 0.1m

= 15 Nm

1. What should be the length of the spanner to loosen the same nut with a 60 N force?

Solution:      

F = 60 N

Τ = 15 Nm

L = ?

$L = \frac{T}{F}$

$L = \frac{15}{60}$

= 0.25 m

2. How much force would be sufficient to loosen it with a 6 cm long spanner?

Solution:      

L = 6 cm = $\frac{6}{100}=0.06m$

Τ = 15 Nm

F =?

F= $\frac{T}{L}$

F= $\frac{15}{0.06}=250N$

Principle of moments:

According to the principle of moments

A body is balanced if the sum of clockwise moments acting on the body is equal to the sum of anticlockwise moments acting on it.

Explanation:

Clockwise moment:

A force that turns a spanner in the clockwise direction is generally used to tighten a nut. The torque or moment of force so produced is called the clockwise moment.

Anticlockwise moment:

On the other hand, to loosen a nut, the force is applied such that it turns the nut in the anticlockwise direction. The torque or moment of force so produced is called anticlockwise moment.

Note:

A body initially at rest does not rotate if the sum of all the clockwise moments acting on it is balanced by the sum of all the anticlockwise moments acting on it. This is known as the principle of moments.

QUICK QUIZ

1. Can a small child play with a fat child on the seesaw? Explain how?

Ans: Yes, they can play on the see-saw, the fat child has a larger weight that’s mean

Larger force and smaller children have a smaller weight and smaller force. So, to play, a larger weight should be a smaller distance from the center of the see-saw and the smaller weight should be at a larger distance from the center of the see-saw. IN another situation a fat child cannot play with a small child if they have equal distances from the center see-saw.

2. Two children are sitting on the see-saw, such that they cannot swing. What is the net torque in this situation?

Ans: Net torque in this situation is zero. Because clockwise torque will cancel the effect of anticlockwise torque.

Center of mass:

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

Explanation:

            It is observed that the center of mass of a system moves as if its entire mass is confined at that point. A force applied at such a point in the body does not produce any torque in it i.e. the body moves in the direction of net force F without rotation.

Center of gravity:

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

Note:

            It is useful to know the location of the center of gravity of a body in problems dealing with equilibrium.

Center of gravity of symmetrical objects:

The center of gravity of objects which have symmetrical shapes can be found in their geometry. 

The center of gravity of a uniform rod:

The center of gravity of a uniform rod lies at a point where it is balanced. This balance point is its middle point G.

Center of gravity of a uniform square or a rectangular sheet:

The center of gravity of a uniform square or a rectangular sheet is the point of intersection of its diagonals.

Center of gravity of a uniform circular disc:

The center of gravity of a uniform circular disc is its center.

Center of gravity of a solid sphere or hollow sphere:

The center of gravity of a solid sphere or hollow sphere is the center of the spheres.

Center of gravity of a uniform circular ring:

The center of gravity of a uniform circular ring is the center of the ring.

Center of gravity of a uniform solid or hollow cylinder:

The center of gravity of a uniform solid or a hollow cylinder is the middle point on its axis.

A simple method to find the center of gravity of a body is by the use of a plumb line.

Plumb line:

A plumb line consists of a small metal bob (lead or glass) supported by a string. When the bob is suspended freely by the string, it rests along the vertical direction due to its weight acting vertically downward. In this state, the center of gravity of the bob is exactly below its point of suspension.

Experiment:

Take an irregular piece of cardboard. Make holes A, B, and C near its edge. Fix a nail on a wall. Support the cardboard on the nail through one of the holes (let it be A), so that the cardboard can swing freely about A. The cardboard will come to rest with its center of gravity just vertically below the nail. A vertical line from A can be located using a plumb line hung from the nail. Mark the line on the cardboard behind the plumb line.

Repeat it by supporting the cardboard from hole B. The line from B will intersect at a point G. Similarly, draw another line from the whole C. Note that this line also passes through G. it will be found that all the vertical lines from holes A, B, and C have a common point G. This common point G is the center of gravity of the cardboard.

Couple:

A couple is formed by two unlike parallel forces of the same magnitude but not along the same line.

Role of a couple in the steering wheel:

When a driver turns a vehicle, he applies forces that produce torque. This torque turns the steering wheel. These forces act on opposite sides of the steering wheel and are equal in magnitude but opposite in direction. These two forces form a couple.

Role of a couple in double arm spanner:

A double arm spanner I used to open a nut. Equal forces each of magnitude F are applied on the ends A and B of a spanner in opposite direction. These forces form a couple that turns the spanner about point O. The torques produced by both the forces of a couple have the same direction. Thus, the total torque produced by the couple will be

Total torque of the couple = F × OA × F × OB

                                          = F(OA + OB)

Torque of the couple   = F × AB       …….  (i)

Equation (i) gives the torque produced by a couple of forces F and F separated by distance AB

Torque of a couple:

The torque of a couple is given by the product of one of the two forces and the perpendicular distance between them.

DO YOU KNOW?

A cyclist pushes the pedals of a bicycle. This forms a couple that acts on the pedals. The pedals cause the toothed wheel to turn to make the rear wheel of the bicycle rotate.

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

Case1:

The first condition for equilibrium does not ensure that a body is in equilibrium.

Consider a body pulled by the forces F1 and F2. The two forces are equal but opposite to each other. Both are acting along the same line; hence, their results will be zero. According to the first condition, the body will be in equilibrium.

Case2:

Now shift the location of the forces as shown in the figure. In this situation, the body is not in equilibrium although the first condition of equilibrium is still satisfied.it is because the body tends to rotate. This situation demands another condition for equilibrium in addition to the first condition i.e. second condition of equilibrium. According to this, a body satisfies the second condition when the resultant torque acting on it is zero.

∑  τ = 0

A paratrooper comes down with terminal velocity and is in equilibrium.

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

Terminal velocity:

The maximum and constant velocity of an object falling vertically downward is called terminal velocity.

Terminal velocity = $V_{t}=2rg^{2}\rho/9 \: \eta$

Where g = acceleration due to gravity, r = radius, ρ = density, η = viscosity.

QUICK QUIZ

1. A ladder leaning against a wall as shown in the figure is in equilibrium. How?

Ans: In this case three forces involved are:

• The weight of the ladder
• The reaction at the wall (R₁)-at right angles because the wall is smooth.
• The reaction at the ground (R₂)-not at a right angle

As the ground is rough and all the forces pass through the same point. The vector diagram for the three forces will cancel the effect of each other therefore ladder leaning at a wall will be in equilibrium.

1. The weight of the ladder in the figure produces an anticlockwise torque. The wall pushes the ladder at its top end thus producing a clockwise torque. Does the ladder satisfy the second condition for equilibrium?

Ans:  Yes, the ladder satisfies the second condition for equilibrium because the clockwise torque will cancel the effect of anticlockwise torque. So, the resultant torque acting in this situation is zero.

1. Does the speed of a ceiling fan go on increasing all the time?

Ans: No, the speed of a ceiling fan does not go on increasing all the time. The fan will move with constant speed.

2. Does the fan satisfy the second condition for equilibrium when rotating with uniform speed?

Ans: Yes, a rotating ceiling fan satisfies the second condition for equilibrium. Because ceiling fan rotating at constant speed is in equilibrium as net torque acting on it is zero

                                                      ∑ τ = 0

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.

Stability and position of the center of mass:

The position of the center of mass of an object plays an important role in its stability. To make them stable, their center of mass must be kept as low as possible.

Examples:

1. Height of vehicles (racing cars) is kept low:

It is due to this reason; that racing cars are made heavy at the bottom and their height is kept to be minimum.

2. Walking of circus artists on a tight rope:

Circus artists such as tight rope walkers use long poles to lower their center of mass. In this way, they are prevented from toppling over.

3. Sewing needle fixed in a cork:

The figure shows a sewing needle fixed in a cork. The cork is balanced on the tip of the needle by hanging forks. The forks lower the center of mass of the system.

1. Perch parrot:

Figure (a) shows a perched parrot that is made heavy at its tail. Figure (b) shows a toy that keeps itself upright when tilted. It has a heavy semi-spherical base. When it is tilted, its center of mass rises. It returns to its upright position at which its center of mass is at its lowest.

A vehicle is made heavy at its bottom to keep its center of gravity as low as possible. A lower center of gravity keeps it stable. Moreover, the base of a vehicle is made wide so that the vertical line passing through its center of gravity should not get out of its base during a turn.