In this article, we shall study two types of products of vectors: a) Scalar product and b) Vector product

Scalar Product of Two Vectors:

The scalar or dot product of two vectors is defined as the product of magnitudes of the two vectors and the cosine of the angles between them.

If  a and  b are two vectors and θ is the angle between the two vectors then by the definition scalar product of two vectors

a · b = a b cos θ

Where  a = magnitude of a and b = magnitude of vector  b

Characteristics of the Scalar Product:

• The scalar product of two vectors is always a pure number i.e. the scalar product is always a scalar.
• The scalar product of two vectors is commutative. i.e. a · b =  b · a

a · b = a b cos θ = b a cos θ = b · a

• Scalar product obeys the distributive law of multiplication. i.e.  a ·( b +  c) = a · b  +  a · c
•  a and b  are two vectors perpendicular to each other, if and only if   a · b =  b · a  = 0
•  a and b  are two vectors parallel to each other, if and only if   a · b =  b · a  = ab
• If  a =  a_x i + a_y j + a_z k and b = b_x i + b_y j + b_z k are the two vectors, then their scalar product is given by

  a · b =(a_x b_x + a_y b_y + a_z b_z )

• The scalar product of two vectors may be zero or positive or negative.

Scalar Products of Standard Unit Vectors:

i · i = i × i cos 0° = 1 × 1× 1 = 1, Similarly,  We have j · j = 1  and  k · k = 1

i · j = i · j = i × j cos 90° = 1 × 1× 0 = 0, Similarly, We have j · k = k · j = o and k · i = i · k = 0

Expression for Scalar Product of Two Vectors:

If  a =  a_x i + a_y j + a_z k and b = b_x i + b_y j + b_z k are the two vectors, then the scalar product is given by

a · b  = ( a_x i + a_y j + a_z k ) · (b_x i + b_y j + b_z k )

∴ a · b  =  a_x i · (b_x i + b_y j + b_z k ) + a_y j · (b_x i + b_y j + b_z k )  + a_z k · (b_x i + b_y j + b_z k )

∴ a · b  =  a_xb_x  i ·i + a_xb_y  i ·j +  a_xb_z   i ·k  +  a_y b_x  j ·i+ a_y b_y j ·j + a_y b_z j ·k

+ a_zb_x  k ·i + a_zb_y k ·j   +  a_zb_z k ·k

∴ a · b  =  a_xb_x (1) + a_xb_y (0) +  a_xb_z (0) +  a_y b_x(0) +  a_y b_y (1) + a_y b_z (0)

+ a_zb_x  (0) + a_zb_y (0)  +  a_zb_z (1)

∴ a · b  =  a_xb_x  +  a_y b_y   +  a_zb_z 

This is an expression for scalar product of two vectors.

Examples of Scalar Product of Two Vectors:

• Work done is defined as scalar product as W = F · s, Where F is a force and s  is a displacement produced by the force
• Power is defined as a scalar product as P = F · v, Where F  is a force and v  is a velocity.

Notes:

• if two vectors are perpendicular to each other then θ = 90° , thus cos θ  = cos 90° = 0 Hence  a · b  =  ab cos 90° = ab(0) = 0
• if two vectors are parallel  to each other then θ = 0° , thus cos θ  = cos 0° = 1 Hence a · b = ab cos 0° = ab(1) = ab
• if two vectors are equal  to each other then θ = 0° , thus cos θ  = cos 0° = 1 Hence  a · a  =  aa cos 0° = aa(1) = a²

Vector Product of Two Vectors:

The vector or cross product of two vectors is a vector whose magnitude is equal to the product of the magnitudes of the two vectors and the sine of the angle between the two vectors.

If a and  b are two vectors and θ is the angle between the two vectors then by the definition  of the vector  product of two vectors

a × b = a b sin θ n

Where  a = magnitude of a and b = magnitude of vector  b

n = unit vector perpendicular to the plane of a and  b

The direction of n is given by right-hand thumb rule

Characteristics of the Vector product:

• Vector product two vectors is always a  vector.
• The Vector product of two vectors is noncommutative. i.e.  a · b ≠  b · a but a · b = –  b · a
• vector product obeys the distributive law of multiplication. i.e.  a ×( b +  c) = a  ×  b  +  a ×  c
• If a · b = 0 and a ≠ o, b ≠ o then the two vectors are parallel to each other.
• If  a =  a_x i + a_y j + a_z k and b = b_x i + b_y j + b_z k are the two vectors, then their vector product is given by

Direction of Vector Product By Right Hand Thumb or Grip Rule:

Hold the right hand along the first vector such that the fingers are parallel to the plane of the vectors and the curled fingers are along the angular direction in which we have to move to the second vector then the outstretched thumb indicates the direction of the vector obtained by vector product of two vectors.

Vector Products of Standard Unit Vectors:

i × i = i × i sin 0° = 1 × 1× 0 = 1, Similarly,  We have j × j = 0  and  k × k = 0

Using the reference circle for vector product

i × j = k ,  j × i = – k  ; j × k = i ,  k × j = – i  ;  k × i = j ,  i × k = – j  ;

Expression for the Vector Product of Two Vectors:

If  a =  a_x i + a_y j + a_z k and b = b_x i + b_y j + b_z k are the two vectors, then the scalar product is given by

a × b  = ( a_x i + a_y j + a_z k ) × (b_x i + b_y j + b_z k )

∴ a × b  =  a_x i × (b_x i + b_y j + b_z k ) + a_y j × (b_x i + b_y j + b_z k )  + a_z k × (b_x i + b_y j + b_z k )

∴ a × b  =  a_xb_x  i × i + a_xb_y  i × j +  a_xb_z   i × k  +  a_y b_x  j × i+ a_y b_y j × j + a_y b_z j × k

+ a_zb_x  k × i + a_zb_y k × j   +  a_zb_z k × k

∴ a × b  =  a_xb_x (0) + a_xb_y (k) +  a_xb_z (- j ) +  a_y b_x(- k) +  a_y b_y (0) + a_y b_z (i)

+ a_zb_x  (j) + a_zb_y (- i)  +  a_zb_z (0)

∴ a × b  =  a_xb_y k  –  a_xb_z j –  a_y b_xk +  a_y b_z i + a_zb_x  j – a_zb_y i

∴ a × b  =  a_y b_z i  – a_zb_y i  –  a_xb_z j + a_zb_x  j    +  a_xb_y k   –  a_y b_xk

∴ a × b  =  i( a_y b_z – a_zb_y) –   j(a_xb_z + a_zb_x)    +  k(a_xb_y    –  a_y b_x )

This is an expression for vector product of two vectors.

Examples of Vector Product of Two Vectors:

• Torque acting on the rotating body is defined as vector product τ = r × F, Where F is a force and r  is a position vector of the point of action of the force.
• If a and b are the adjacent sides of a parallelogram, then the area of the parallelogram is given as A =  | a × b|
• If a and b are the adjacent sides of a triangle, then the area of the triangle is given as A =  ½| a × b|

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In this article, we shall study the concept of moment of force, its characteristics, and applications in everyday use.

Rigid Body:

A rigid body is one whose geometric shape and size remains unchanged under the action of an external force. In the rigid body, the distance between any two particles of the body remains constant. In other words the relative position of each particle w.r.t. any other particle always remains the same. If force is applied (whatever may be its magnitude) there is no change in the shape of the body. Although no perfectly rigid body exists, many bodies can be considered as rigid bodies for a practical purpose.

Rotational Motion and Axis of Rotation:

A rigid body is said to perform rotational motion when the particles lying on the straight line in the body remains stationary and all other particles move in circles around this straight line. The straight line inside the body which remains fixed is called the axis of rotation.

When a force is applied on a rigid body which is free to move, the body starts moving in a straight line in the direction of the force. The motion of the body is called linear or translational motion. But if the body is pivoted at a point, the force applied on the body at suitable point rotates the body about its fixed point.(or about the axis passing through the fixed point) This is called rotational motion. For example, when a force is applied at the handle of the door, the door rotates.

A single force can produce translational motion of a body if it is free to move, but a single force applied on a body fixed at a point does not cause rotational motion of the body.

Moment of Force:

In translational motion the linear acceleration of body depends on the force acting on the body, similarly, in rotational motion, angular acceleration depends on the moment of force or torque acting on the body. The ability of a force to produce rotational motion is measured in terms of the moment of force. Its magnitude about the given axis depends on the magnitude of the force and the perpendicular distance of the line of action of force from the axis of rotation. This distance is called the moment arm.

The Magnitude of moment = force × moment arm

It is denoted by letter M. Its S.I. unit is N.m. Its dimensions are [m¹L²T^-2]. It is a vector quantity, whose direction is given by right-hand thumb rule.

Sign Convention Used for Moment of Force:

• If the force applied to the body rotates the body in an anticlockwise sense, then it is considered as positive.
• If the force applied to the body rotates the body in a clockwise sense, then it is considered as negative.
• The direction of the moment of force (torque) is given by right-hand rule. It states that “Encircle the axis of rotation by fingers of the right hand which point in the direction in which the body tends to rotate, then the thumb points in the direction of torque or moment of the force vector.

Expression for Moment of Force:

Consider a body of any shape capable of rotating about an axis passing through point O and perpendicular to the plane of paper as shown in the diagram

Let P be a point in the plane the f paper.  Let r  be the position vector of point P with respect to point O. Let F be the force acting at point P making an angle θ with position vector . In right-angled Δ PQO,

sin θ = OQ/OP

OQ = OP Sin θ = r Sin θ

The magnitude of the moment of force = Force  ×   Moment arm 

Thus the moment of force is a vector product of moment arm and the force. Its direction is given by right-hand thumb rule. In this case, the direction is perpendicular to the plane of the paper and towards us.

Scientific Reasons:

It is easy to open a door by applying the force at the free end.

The door rotates about an axis passing through its hinges.

Required moment of force to rotate the door = force x moment arm

When we apply force at the free end, the distance of the force from the axis of rotation (moment arm) is more. Thus less force is required to open the door due to a large turning effect.

The hand flour grinder is provided with a handle near its rim.

The upper stone of a flour grinder rotates about an axis passing through its centre.

Required moment of force to rotate the stone = force x moment arm

When the handle is near the rim, the distance of the force from the axis of rotation (moment arm) is more. Thus less force is required to rotate the stone of a flour grinder is less due to large turning effect.

A long handle spanner is used to loosen or tight nut.

During loosening or tightening the nut rotates about an axis passing through its centre.

Required moment of force to rotate the spanner = force x moment arm

When long handle spanner is used, the distance of the force from the axis of rotation (moment arm) is more. Thus less force is required to loosen or tight the nut due to large turning effect.

In bicycle long pedals are used.

In bicycle, the pedals are used to rotate the toothed wheel about an axis passing through its centre.

Required moment of force to rotate the wheel = force x moment arm

When long pedals are used, the distance of the force from the axis of rotation (moment arm) is more. Thus less force is required to rotate the toothed wheel due to large turning effect.

Principle of Moments:

If a body is in rotational equilibrium then the sum of the anticlockwise moments is equal to the sum of the clockwise moments. OR If a body is in rotational equilibrium then the algebraic sum of the moments about any point is zero.

Applications of Principle of Moments:

• To find the mass of an object
• In levers (Simple Machine)

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In this article, we shall study the concept of a couple, its characteristics, and applications.

Parallel Forces:

Forces which are not concurrent and their lines of action are not the same and parallel to each other then the forces are called parallel forces.

Example: Children sitting on a see-saw. Their weights are the example of parallel forces.

If there are two forces which are equal and opposite they form a couple.

Couple:

When two forces of equal magnitude opposite in direction and acting along parallel straight lines, then they are said to form a couple. The perpendicular distance between the two force forming a couple is called the arm of the couple.

Effects of Couple:

Due to the action of a couple, a torque acts on a body which makes the body to rotate about a fixed point. The torque acting on the body is given by

The magnitude of torque = Magnitude of force × perpendicular distance between the two forces.

Characteristics of the Couple:

• As the two forces forming couple are equal and opposite the couple doesn’t produce translational motion.
• When it acts on a body net resultant force acting on a body is zero.
• It produces pure rotational motion in the body this is because the algebraic sum of the moments of the two forces about any point in their plane is not zero.
• The moment of a couple about any point in its plane is constant, both in magnitude and direction.

Practical Application of a Couple:

Couple is required

• to turn a car the driver applies a couple to the steering wheel
• to wind the spring of an alarm clock, it  is applied by the fingers
• to open or close a water tap
• to turn a key of a lock.
• to open or close a cap of a bottle.
• to turn a screwdriver.

Moment of a Couple:

The ability of a couple to produce rotational motion in a body is measured in terms of torque. The magnitude of the torque is equal to the product of one of the forces and the perpendicular distance between the lines of action of the two forces. The moment of a couple about any point in its plane is constant and is equal to the product of one of the forces and the perpendicular distance between the lines of action of the two forces.

Moment of a Couple About Any Point in its Plane is Constant:

Case – I: When Point O is between A and B

Taking moment about O

τ = F . O B × F . O A

τ = F .(OB + OB)

τ= F. (AB)

Case – II: When Point A is between O and B

Taking moment about O.

τ = – F . O A + F . O B

τ = F.(AB)

Case – III: When Point B is between O and A

Taking moment about O.

τ = – F . OB + F . O A

τ = F . (OA – OB)

τ = F. (AB)

In each case, the moment of couple or torque has the same magnitude. It can be proved for any point in the plane of the forces. This proves that the moment of the couple about any point in its plane is constant and is equal to the product of one of the forces and the perpendicular distance between the lines of action of the two forces.

Notes:

A couple or torque produces only the rotational motion of a body.

Let us assume that a couple is applied to a body at rest. The forces constituting the couple are equal in magnitude but opposite in direction. Therefore their resultant is zero.  As there is no resultant force acting on the body, the body does not have translational motion. However, the moment of the couple about any point in its plane is not equal to zero. As a result of this moment, the body is set into purely rotational motion.

Torque and work have the same dimensions but they are different physical quantities.

Work and moment of force (torque) each are the product of force and distance. Hence their dimensions are the same. The dimensions of the moment of force (torque) are [M¹L²T-2] which are the same as work.

When calculating work, the force and the distance are measured in the same direction while the calculating moment of force (torque) the force and the distance are measured in the direction perpendicular to each other.

Moment of force is vector quantity while the work in scalar quantity. Hence though the torque and work have the same dimensions, they are different physical quantities.

The moment of force changes with the change in the axis of rotation while the moment of a couple about any axis is constant.

The magnitude of the moment of a force about the given axis depends on the magnitude of the force and the perpendicular distance of the line of action of a force from the axis of rotation. If the position of the axis of rotation changes the distance of the force from the axis of rotation changes. Hence the moment of force changes.

The magnitude of the moment of couple or torque is equal to the product of one of the forces and the perpendicular distance between the lines of action of the two forces. The calculation is not linked with the axis of rotation. Hence it is independent of the position of the axis of rotation. Hence its magnitude remains constant, even if the axis is changed.

Equilibrium of a Body:

For a body to be in translational equilibrium the resultant force acting on a body should be zero. In such an equilibrium, the body has no translation motion.

∴   ∑ F = 0

For a body to be in rotational equilibrium the resultant moment acting on a body should be zero. In this equilibrium, the body has no rotational motion.

∴   ∑ M = 0

For a body in total equilibrium, the body neither has translational motion nor has rotational motion.

i.e ∑ F = 0 and ∑ M = 0

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