Table of Contents
Notes
Q1. Can the nut of the axle of a bike be loosened with hand why we use a spanner for this purpose?
Ans: 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.
Q2. What is the joker doing in the figure?
Ans: He is trying to balance himself on a wooden plank which is placed over a cylindrical pipe. Due to open the arms he is doing its center of mass as low as possible to make him stable.
Q3. Women and children in the villages often carry pitchers with water on their heads how this is possible?
Ans: Woman and children keep itself upright when carrying pitchers on their heads. The pitcher has a heavy semispherical
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.
Q4. With a little effort, we can learn to balance a stick vertically up on our fingertip how this is possible?
Ans: 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.
Q5. What is meant by parallel forces?
Ans: 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.
Q6. What is the difference between like and unlike parallel forces?
OR
Define like and unlike parallel force?
Ans: See Q # 4.3(i) from Exercise.
Q7. Many people push a bus to start it why all of them push it in the same direction?
Ans: Like forces acting in the same direction increases the resultant force which moves the bus easily.
Q8. Explain the unlike parallel forces in the given figure?
Ans: An apple is suspended by a string. The string is stretched due to 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, unlike parallel forces.
Q9. Explain the unlike parallel forces in the given figure?
Ans: In the figure, Forces F_{1 }and F_{2 }are also unlike parallel Forces, because they are parallel and opposite to each
Other. But F_{1} and F_{2} are not acting along the same line and Hence, they are capable to rotate the body.
Q10. Define the resultant vector?
Ans: 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 vector 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.
Q11. How head to tail rule helps to find resultant of forces?
Ans: See Q # 4.4 from Exercise
Q12. What is meant by trigonometry? Give some important trigonometric ratios.
Ans: Trigonometry:
Trigonometry is that branch of mathematics which deals with the properties of a rightangled triangle.
Trigonometric ratios:
Consider a rightangled triangle ∆ABC having θ at A.
sin θ = Perpendicular/ Hypotenuse = BC/AB
cos θ = Base/Hypotenuse = AC/AB
tan θ = Perpendicular/Base = BC/AC
Note:
To remember trigonometric ratios, we use following sentence:
“Some people have – Curly brown hair – Through proper brushing”
Pythagoras theorem:
(Hypotenuse)^{2 }= (Base)^{2} + (Perpendicular)^{2}
^{ }
Q13. How can a force be resolved into its rectangular components?
OR
Explain the resolution of the vector?
Ans: See Q # 4.5 from Exercise
Trigonometric Table
Ratio/θ  0⁰  30⁰  45⁰  60⁰  90⁰ 
sin θ  0  0.5  0.707  0.866  1 
cos θ  1  0.866  0.707  0.5  0 
tan θ  0  0.577  1  1.732  ∞ 
Mini Exercise
In a rightangled 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:
 Length of hypotenuse:
Pythagoras theorem:
(Hypotenuse)^{2 }= (Base)^{2} + (Perpendicular)^{2}
(Hypotenuse)^{2 }= (4)^{2} + (3)^{2}
(Hypotenuse)^{2 }= 16 + 9
(Hypotenuse)^{2 }= 25 by taking square root on both sides
Hypotenuse = 5 cm
 sin θ:
sin θ = Perpendicular / Hypotenuse = 3 / 5
 cos θ:
cos θ = Base / Hypotenuse = 4 / 5
 tan θ:
tan θ = Perpendicular / Base = 3 / 4
Q14. Briefly explain the determination of a force from its perpendicular components?
Ans: 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 tail rule:
OR = OP + PR
Thus, OR will completely represent the force F whose x and ycomponents are F_{X} and F_{y} respectively. That is
F = F_{X} + F_{y}
Magnitude of resultant force/Magnitude of resultant vector:
The magnitude of the force F can be determined using the right angled triangle OPR
As
(OR)^{2} = (OP)^{2} + (PR)^{2}
F^{2} = F_{X}^{2} + F_{y}^{2}
Hence
F = √F_{x}^{2} + F_{y}^{2} (i)
Direction of the resultant force/Direction of the resultant vector:
The direction of the force F with xaxis is given by
tan θ = PR / OP = F_{y} / F_{X}
θ = tan^{1} F_{y} / F_{X}
_{}
_{ }
Q15. Why it is easy to open and close the door by pulling or pushing it as it handles?
Ans: We open or close a door by pushing or pulling it. Here push or pull 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.
Q16. What do you mean by a rigid body?
Ans: 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 the one that is not deformed by force or forces acting on it.
Q17. What do you mean by the axis of rotation?
Ans: 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.
Q18. Name some objects that work by the turning effects of forces.
Ans: Turning pencil in sharpener, turning stopcock of a water tap, turning the doorknob and so on are some of the examples where a force produces turning effect.
QUICK QUIZ

Name some more objects that work by the turning effects of forces.
Ans: (i) Torque is produced when a force is applied to 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.
Q19. Define torque. What is its unit? On what factors torque (moment of a force) depends?
Ans: Torque (moment of a force):
The turning effect of a force is called torque or moment of the force.
Torque τ = F × L
Torque is a vector quantity and its direction can be found by using the righthand rule.
Unit of torque:
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.
 Magnitude of the force(F)
Greater is a force, greater is the moment of the force.
Τ ∝ F ………………………………………. (i)
 Moment arm
Similarly, longer is the moment arm, greater is the moment of the force.
Τ ∝ L ………………………………………. (ii)
Q20. Why the handle of a door is fixed near the outer edge of a door?
OR
Why door handles usually on the opposite edge of the door from the hinge?
Ans: 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 of a door. (Τ ∝ L)
Q21. Why it is easy to tighten a nut using a spanner of the longer arm than a spanner of the shorter arm?
Ans: A spanner having long arm helps to loosen or tighten a nut or a bolt with greater ease than the one having short arm. It is because of the turning effect(torque) of the force increases. (Τ ∝ L)
Q22. What do you mean by a line of action of a force?
Ans: Line of action of a force:
The line along which a force act is called the line of action of the force. In figure, line BC is the lie of action of force F.
Q23. Define the moment arm.
Ans: 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.
Q24. What do you mean by newtonmeter (Nm)?
Ans: SI unit of torque is newtonmeter (Nm).
Newtonmeter (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 spanner 10 cm long.
Solution: F = 150 N
L = 10 cm = 10 / 100 = 0.1 m
Torque Τ = F × L
= 150 N × 0.1 m
= 15 Nm

What should be the length of the spanner to loosen the same nut with a 60 N force?
Ans: F = 60 N
Τ = 15 Nm
L = ?
L = Τ / F
L = 15 / 60
= 0.25 m

How much force would be sufficient to loosen it with a 6 cm long spanner?
Solution: L = 6 cm = 6 /100 = 0.06 m
Τ = 15 Nm
F =?
F = Τ / L
F = 15 / 0.06 = 250 N
Q25. Describe the principle of the moment?
Ans: 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 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

Can a small child play with a fat child on the seesaw? Explain how?
Ans: Yes, they can play on seesaw, the fat child has a larger weight that’s mean
Larger force and smaller child 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 larger distance from the center of the see saw. IN another situation a fat child cannot play with small child if they have equal distances from the center seesaw.

Two children are sitting on the seesaw, 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.
Q26. Explain how center of mass helps the system to move as well as rotate?
Ans: Center of mass:
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.
Q27. Define center of gravity?
Ans: Center of gravity:
A point where the whole weight of the body appears to act vertically downward is called center of gravity of a body.
Note:
It is useful to know the location of center of gravity of a body in problems dealing with equilibrium.
Q28. List the center of gravity of some symmetrical objects?
Ans: Center of gravity of symmetrical objects:
The center of gravity of objects which have symmetrical shapes can be found from 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.
No.  Object  Center of gravity 
1.  Uniform rod  Center of the rod 
2.  Round plate  Center of the plate 
3.  Sphere  Center of the sphere 
4.  Triangular plate  Point of intersection of the medians 
5.  Cylinder  Central point of axis 
6.  Square, Rectangle, parallelogram  Point of intersection of the diagonals 
Q29. Explain an experiment to find the center of gravity of a foursided plate of uniform thickness. How can you verify your answer by using geometry?
OR
Explain an experiment to find the center of gravity of an irregular shaped thin lamina?
Ans: 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, 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 the 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. I 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.
Q30. Define a couple. Describe its role in steering wheel double arm spanner?
Ans: 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 a 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 to rotate.
Q31. When a body is said to be in equilibrium?
OR
Define equilibrium.
Ans: See Q # 4.6 from Exercise.
Q32. Explain the first condition for equilibrium.
Ans: See Q # 4.7 from Exercise.
Q33. What is the second condition for equilibrium?
Ans: See Q # 4.9 from Exercise.
Q34. Why there is a need for second condition for equilibrium if a body satisfies the first condition for equilibrium?
Ans: See Q # 4.8 from Exercise.
Q35. How does a paratrooper come down?
Ans: A paratrooper comes down with terminal velocity is in equilibrium.
A paratrooper coming down with terminal velocity (constant velocity) also satisfies the first condition for equilibrium and is thus in equilibrium.
Q36. Define terminal velocity?
Ans: Terminal velocity:
The maximum and constant velocity of an object falling vertically downward is called terminal velocity.
Terminal velocity = V_{t }= 2gr^{2 }ρ / 9 η
Where g = acceleration due to gravity, r = radius, ρ = density, η = viscosity.
QUICK QUIZ

A ladder leaning against at 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 (R_{1})at right angles because the wall is smooth.
 The reaction at the ground (R_{2})not at 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.

The weight of the ladder in the figure produces an anticlockwise torque. The wall pushes the ladder at its top end thus produces 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.

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. Fan will move with constant speed.

Does the fan satisfy second condition for equilibrium when rotating with uniform speed?
Ans: Yes, a rotating ceiling fan satisfies second condition for equilibrium. Because ceiling fan rotating at constant speed is in equilibrium as net torque acting on it is zero
∑ τ = 0
Q37.Explain what is meant by stable, unstable and neutral equilibrium. Give one example in each case.
OR
Briefly explain the states of equilibrium?
Ans: States of equilibrium:
There are three states of equilibrium: stable equilibrium, unstable equilibrium and neutral equilibrium.
 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 sets 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.
 Unstable equilibrium:
If a body does not return to its previous position when sets 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 the state of unstable equilibrium.
Features of unstable equilibrium:
The center of gravity of the body is at its highest position in the state of unstable equilibrium. As the body topples over about its base (tip), it’s 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 increases their stability.
 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 from 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 to its new position.
Q38. Discuss the stability and position of center of mass with the reference of example?
OR
Give a few examples in which lowering of center of mass make the objects stable?
Ans: Stability and position of center of mass:
Position of center of mass of an object plays an important role in their stability. To make them stable, their center of mass must be kept as low as possible.
Examples:
 Height of vehicles (racing car) is kept low:
It is due to this reason; racing cars are made heavy at the bottom and their height is kept to be minimum.
 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.
 Sewing needle fixed in a cork:
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.
 Perch parrot:
Figure (a) shows a perched parrot which is made heavy at its tail. Figure (b) shows a toy that keeps itself upright when tilted. It has a heavy semispherical 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.
Q39. Why a vehicle is made heavy at its bottom?
Ans: 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.
Summary
 Parallel forces: Parallel forces have their lines of action parallel to each other.
 Like Parallel forces: If the direction of parallel forces is the same, they are called parallel forces.
Unlike Parallel forces: If two parallel forces are in opposite direction to each other then they are called unlike parallel forces.
 Resultant force: The sum of two or more forces is called the resultant force.
 Head to the tail rule: A graphical method used to find the resultant of two or more forces is called head to tail rule.
 Resolution of that force: Splitting up a force into two components perpendicular to each other is called resolution of that force. These components are:
F_{x} = F cos θ F_{y }= F sin θ
 A force can be determined from its perpendicular components as
F = √F_{x}^{2} + F_{y}^{2}
 Torque or moment: Torque or moment of the force is the turning effect of the force. Torque of a force is equal to the product of the force and moment arm of the force.
 Principle of the moment: According to the principle of moments, the sum of clockwise moments acting on a body in equilibrium is equal to the sum of the anticlockwise moments acting on it.
 Centre of mass: Centre of the mass of a body is such a point where the net force causes it to move without rotation.
 Centre of gravity: The center of gravity of a body is a point where the whole weight of a body acts vertically downward.
 A couple is formed by two parallel forces of the same magnitude but acting in opposite direction along different lines of action.
 A body is in equilibrium if net force acting on it is zero. A body in equilibrium either remains at rest or moves with a uniform velocity.
 A body is said to satisfy second condition for equilibrium if the resultant torque acting on it is zero.
 A body is said to be in stable equilibrium if after a slight tilt it returns to its previous position.
 Unstable equilibrium: If a body does not return to its previous position when sets free after slightly tilt is said to be in unstable equilibrium.
 Neutral equilibrium: A body that remains in its new position when disturbed from its previous position is said to be in a state of neutral equilibrium.
what will happen when we increase the length of spanner
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