Identify the correct order in which the value of normal reaction increases, (object is placed on rough horizontal surface)
i) The object is pushed with the force F at an angle 'q' with horizontal
ii) The object is pulled with the force F at an angle 'q' with horizontal
iii) The object is pushed down with the force F normally
iv) The object pulled up with the force F normally

Options

A. i, ii, iii, iv
B. iv, ii, i, iii
C. i, iii, ii, iv
D. ii, iv, iii, i

Answer

To solve the problem of determining the correct order in which the value of normal reaction increases when an object is placed on a rough horizontal surface and subjected to different forces, we will analyze each case step by step.

### Step 1: Understand the Forces Acting on the Object
For an object of mass \( M \) placed on a horizontal surface, the forces acting on it include:
- The weight of the object, \( Mg \), acting downwards.
- The normal reaction force, \( N \), acting upwards.
- Any applied force \( F \) at an angle \( \theta \) with respect to the horizontal.

### Step 2: Analyze Each Case

#### Case i: Object is Pushed with Force \( F \) at Angle \( \theta \)
- The vertical component of the force \( F \) is \( F \sin \theta \) acting downwards.
- The normal reaction \( N_1 \) can be expressed as:
\[ N_1 = Mg + F \sin \theta \]

#### Case ii: Object is Pulled with Force \( F \) at Angle \( \theta \)
- The vertical component of the force \( F \) is \( F \sin \theta \) acting upwards.
- The normal reaction \( N_2 \) can be expressed as:
\[ N_2 = Mg - F \sin \theta \]

#### Case iii: Object is Pushed Down with Force \( F \) Normally
- Both the weight \( Mg \) and the force \( F \) act downwards.
- The normal reaction \( N_3 \) can be expressed as:
\[ N_3 = Mg + F \]

#### Case iv: Object is Pulled Up with Force \( F \) Normally
- The weight \( Mg \) acts downwards while the force \( F \) acts upwards.
- The normal reaction \( N_4 \) can be expressed as:
\[ N_4 = Mg - F \]

### Step 3: Compare the Normal Reactions
Now we have the expressions for the normal reactions:
1. \( N_1 = Mg + F \sin \theta \)
2. \( N_2 = Mg - F \sin \theta \)
3. \( N_3 = Mg + F \)
4. \( N_4 = Mg - F \)

To determine the order of these normal reactions, we need to consider the effects of \( F \) and \( \sin \theta \):
- Since \( \sin \theta \) ranges from 0 to 1, \( N_2 \) (which decreases with \( F \sin \theta \)) will be the smallest when \( F \sin \theta \) is maximized.
- \( N_4 \) is also reduced by \( F \), but it is less than \( N_3 \) and \( N_1 \) since both of these add to the weight.

### Step 4: Arrange in Increasing Order
1. **Smallest**: \( N_4 = Mg - F \) (pulled up normally)
2. **Next**: \( N_2 = Mg - F \sin \theta \) (pulled with force at angle)
3. **Next**: \( N_1 = Mg + F \sin \theta \) (pushed with force at angle)
4. **Largest**: \( N_3 = Mg + F \) (pushed down normally)

### Final Order
Thus, the correct order in which the value of normal reaction increases is:
\[ N_4 < N_2 < N_1 < N_3 \]