Unit2 - Subjective Questions

Friction is defined as the opposing force that comes into play when one body moves or tends to move over the surface of another body. It always acts in a direction opposite to the direction of motion or impending motion.

Causes of Friction:

1. Surface Irregularities: No surface is perfectly smooth. At the microscopic level, surfaces possess hills and valleys. When two surfaces are in contact, these irregularities interlock. Overcoming this interlocking requires a force, which manifests as friction.
2. Molecular Adhesion: When two surfaces are in close contact, molecular forces of attraction (adhesion) between the material surfaces act at the points of actual contact (asperities). This creates 'cold welds' which must be sheared to allow motion.

Friction can be broadly classified into two categories based on the nature of the interacting surfaces and their state:

1. Dry Friction (Coulomb Friction): Exists when unlubricated surfaces of two solids are in contact and tend to slide over each other.
2. Fluid Friction: Occurs between layers of fluids moving relative to each other, or between a solid and a fluid layer (e.g., in lubricated bearings).

Dry friction is further classified based on the state of motion:

• Static Friction: The friction experienced by a body when it is at rest. It is a self-adjusting force that ranges from zero to a maximum value.
• Dynamic (Kinetic) Friction: The friction experienced by a body when it is in motion. It is generally less than the maximum static friction.

Dynamic friction is subdivided into:

• Sliding Friction: Experienced when a body slides over another (e.g., a block pushed on a floor).
• Rolling Friction: Experienced when a body rolls over another (e.g., a wheel or roller bearing).

Static Friction:

• Definition: The opposing force present when a body tends to move but is not actually moving.
• Nature: It is a self-adjusting force. It is equal and opposite to the applied force until motion impends.
• Magnitude: Varies from zero to a maximum value called 'Limiting Friction' ().
• Coefficient: Governed by the coefficient of static friction ().

Dynamic (Kinetic) Friction:

• Definition: The opposing force present when a body is in actual relative motion over another surface.
• Nature: It is constant in magnitude and does not depend on the applied force.
• Magnitude: Slightly less than the limiting static friction ().
• Coefficient: Governed by the coefficient of kinetic friction (), where .

Limiting Friction:
It is defined as the maximum value of static friction that comes into play when a body is just on the verge of moving over the surface of another body. When the applied force is less than limiting friction, the body remains at rest. The moment the applied force exceeds this limiting value, the body starts to move.

Significance:

• It represents the threshold between equilibrium (rest) and motion.
• It is used to define the coefficient of static friction, , where is the limiting friction and is the normal reaction.
• Engineers use limiting friction to determine the maximum force a structure or component can withstand before slipping occurs.

The Laws of Static Friction (also known as Coulomb's Laws of Dry Friction) are:

1. The force of friction always acts in a direction opposite to the direction in which the body tends to move.
2. The magnitude of the force of static friction is exactly equal to the applied force until the body is on the verge of moving (self-adjusting nature).
3. The magnitude of limiting friction () bears a constant ratio to the normal reaction () between the two surfaces. i.e., .
4. The force of friction depends on the nature and roughness of the surfaces in contact.
5. The force of friction is independent of the apparent area of contact between the surfaces, provided the normal reaction remains the same.

The Laws of Dynamic or Kinetic Friction are as follows:

1. The force of dynamic friction always acts in a direction opposite to that in which the body is moving.
2. The magnitude of dynamic friction bears a constant ratio to the normal reaction () between the two surfaces. i.e., .
3. For moderate speeds, the force of dynamic friction is practically constant and independent of the velocity of motion.
4. The coefficient of dynamic friction () is slightly less than the coefficient of static friction ().
5. Like static friction, it depends on the nature of the surfaces in contact but is independent of the area of contact.

Definition:
The angle of friction () is defined as the angle made by the resultant of the normal reaction () and the limiting force of friction () with the normal reaction.

Derivation:
Consider a body of weight resting on a horizontal plane. Let be the applied force such that the body is just on the point of sliding.

• Normal reaction =
• Limiting friction =
• Let be the resultant of and .
• Let be the angle between and .

From the geometry of the force triangle (forming a right-angled triangle with , , and ):

By the laws of solid friction, the ratio of limiting friction to normal reaction is the coefficient of friction ().

Therefore, equating the two expressions:

Hence, the coefficient of static friction is equal to the tangent of the angle of friction.

Angle of Repose ():
It is the maximum angle of inclination of a rough inclined plane with the horizontal, at which a body placed on it remains in equilibrium (just on the point of sliding down) without the application of any external force.

Proof:
Consider a body of weight resting on a plane inclined at an angle to the horizontal. The body is on the verge of sliding down.
Forces acting on the body:

1. Weight acting vertically downward.
2. Normal reaction perpendicular to the plane.
3. Frictional force acting upwards along the plane.

Resolving forces parallel to the plane:
--- (Equation 1)

Resolving forces perpendicular to the plane:
--- (Equation 2)

Dividing Equation 1 by Equation 2:

We know that the coefficient of friction and also where is the angle of friction.
Therefore:


Hence, the angle of repose is exactly equal to the angle of friction.

The Cone of Friction is a conceptual geometrical cone used in mechanics to determine if a body remains in equilibrium under a given set of forces.

Explanation:

• When a body rests on a rough surface, the resultant of the normal reaction and the static frictional force makes an angle with the normal.
• When the frictional force is at its limiting value (), this angle becomes the angle of friction .
• If the direction of impending motion is rotated a full 360 degrees on the contact surface, the resultant vector traces out the surface of a cone.
• This cone has its vertex at the point of contact, its axis along the normal reaction , and a semi-vertical angle equal to the angle of friction .
• Significance: As long as the line of action of the resultant contact force falls inside or on the surface of this cone, the body will remain at rest. If the required resultant falls outside the cone, the body will slide. Thus, the cone represents the zone of equilibrium.

Definition:
The coefficient of friction () is defined as the ratio of the limiting force of friction () to the normal reaction () between two bodies in contact.

It is a dimensionless quantity.

Significance:

• It represents the degree of roughness between two surfaces. A higher value indicates rougher surfaces with higher frictional resistance.
• It is vital in engineering design, such as deciding materials for brake pads (high needed) or bearings (low needed).

Properties and Values:

• It depends on the nature of the materials in contact and the condition of the surfaces (dry, wet, polished, lubricated).
• It is independent of the mass of the body or the apparent area of contact.
• The coefficient of static friction () is always greater than the coefficient of kinetic friction ().
• Typical values for dry surfaces range from 0.1 to 1.0 (e.g., steel on steel , ice on ice ).

Let a body of weight rest on a rough horizontal plane. Let be the force applied at an angle above the horizontal to just move the body.

• = Normal reaction
• = Frictional force =

Resolving forces horizontally:

Resolving forces vertically:

Substituting in the horizontal equation:

Since where is the angle of friction:

For to be minimum, the denominator must be maximum.
The maximum value of is 1, which occurs when .
Therefore, the minimum force required is , applied at an angle equal to the angle of friction.

Consider a body of weight moving up a rough inclined plane of inclination . A force is applied parallel to the plane to move the body upwards.
Since the motion is impending upwards, the frictional force acts downwards along the plane.

Forces acting on the body:

1. Weight acting vertically down.
2. Normal reaction perpendicular to the plane.
3. Frictional force acting downwards along the plane.
4. Applied force acting upwards along the plane.

Resolving forces perpendicular to the plane:

Resolving forces parallel to the plane:

Substitute :

Substituting :



This equation gives the force required to move the body up the inclined plane.

Consider a body of weight on a rough plane inclined at angle . Let (angle of repose), so the body will not slide down by itself. A downward force is applied parallel to the plane to pull it down.
Since motion is impending downwards, friction acts upwards.

Forces acting:

1. Weight downwards.
2. Normal reaction perpendicular to plane.
3. Friction acting upwards along the plane.
4. Applied force acting downwards along the plane.

Resolving forces perpendicular to the plane:

Resolving forces parallel to the plane:


Substitute :

Substituting :



This is the force required to pull the body down the plane.

Impending Motion:
Impending motion refers to the state of a body when it is on the verge of slipping or moving, but actual relative motion has not yet begun. At this exact instant, the static frictional force has reached its absolute maximum value, known as limiting friction ().

Before impending motion, the friction is static and self-adjusting (). Once the applied force exceeds the state of impending motion, the body starts to move, and the friction becomes kinetic.

Effect on Direction:
Friction always opposes relative motion. Therefore, the direction of the frictional force is always strictly exactly opposite to the direction of the impending motion. By analyzing the direction the body would move if friction were zero, one can confidently determine the direction in which the limiting frictional force acts.

Coefficient of Friction as an Experimental Constant:
The coefficient of friction () is an empirical value, meaning it is derived from experimental observations rather than being a fundamental law of physics like the gravitational constant.

Reasons:

1. Surface Dependency: It is highly dependent on the microscopic texture, hardness, and material properties of the two specific surfaces in contact.
2. Environmental Factors: It can change drastically with environmental conditions like temperature, humidity, and the presence of dust or contaminants.
3. Wear and Tear: As surfaces wear out over time, their roughness changes, thereby altering the value of .
   Therefore, is a system property (pair of surfaces) rather than a material property, and tables of values are always approximations based on specific controlled test conditions.

Sliding Friction:

• Occurs when two surfaces rub against each other with continuous relative sliding motion (e.g., a sled on snow, pushing a box on the floor).
• It involves the continuous shearing of the microscopic asperities (irregularities) between the surfaces.

Rolling Friction:

• Occurs when a body (like a cylinder, sphere, or wheel) rolls over a surface without slipping.
• The theoretical point of contact changes continuously.

Why Rolling Friction is Less:
In pure rolling, there is ideally no relative slip at the point of contact. The resistance (rolling friction) arises primarily from the elastic deformation of the rolling body and the surface it rolls on. As the wheel rolls, it creates a small depression in the surface and effectively has to 'climb' out of it, which requires a small amount of energy. Because this deformation energy is much smaller than the energy required to shear interlocking microscopic asperities in sliding, rolling friction is significantly less than sliding friction.

Given:

• Mass of block,
• Applied force,
• Acceleration due to gravity,

Step 1: Calculate Weight and Normal Reaction
Weight of the block, .
Since the block is on a horizontal floor, Normal Reaction .

Step 2: Calculate Coefficient of Static Friction ()
The force required to just move the block is equal to the limiting friction .
Therefore, .
We know that

Step 3: Calculate Angle of Friction ()
We know the relation

Final Answer:
Coefficient of static friction is $0.3058$ and the angle of friction is .

Let a block of weight rest on a rough inclined plane with inclination .
Let be the angle of friction and be the coefficient of friction.

The forces acting on the block parallel to the plane:

1. Downward component of weight =
2. Upward frictional force opposing motion =

The normal reaction perpendicular to the plane is .

The maximum possible static friction (limiting friction) is:

For the block to be in equilibrium and just on the verge of sliding down, the downward force must equal the limiting friction:

We also know from the definition of the angle of friction that .
Therefore:

Thus, if the inclination angle equals the angle of friction , the body is in limiting equilibrium solely under its own weight, requiring no external force.

Effect of Material Choice on Friction:
The coefficient of friction is fundamentally a property of the pair of contacting materials. Different materials have different molecular structures, hardness, and surface properties.

• Similar Metals: Often have high friction and a tendency to "gall" or cold-weld together (e.g., aluminum on aluminum).
• Dissimilar Metals: Generally exhibit lower friction (e.g., steel on brass).
• Polymers and Composites: Materials like Teflon (PTFE) or Nylon inherently possess low coefficients of friction against metals, even without lubrication.
• Elastomers: Rubber has a very high coefficient of friction on rough surfaces like concrete (vital for tires).

Importance in Mechanical Design:
Engineers must select materials based on whether friction is desired or detrimental in a specific application:

1. Where friction is a "necessary evil" (To be minimized): Bearings, gears, and piston rings require materials with low (like bronze or babbit metal) to minimize wear, energy loss, and heat generation.
2. Where friction is beneficial (To be maximized): Brakes, clutches, and belt drives require materials with high (like composite brake pads on steel rotors) to transmit force or arrest motion effectively without slipping.

To analyze the motion (or equilibrium) of a body involving dry friction, follow these general steps:

1. Assume the State: Initially, assume the body is in static equilibrium.
2. Free Body Diagram (FBD): Draw the FBD of the body. Include all applied active forces, the weight of the body, the normal reaction , and the static frictional force .
3. Determine Direction of Friction: Determine the direction of impending motion (the direction the body would move if surfaces were perfectly smooth). Draw the friction force exactly opposite to this direction.
4. Apply Equilibrium Equations: Use the equations of statics (, ) to solve for the required frictional force and the normal reaction .
5. Calculate Limiting Friction: Calculate the maximum available static friction using the formula .
6. Compare and Conclude:
   • If : The assumption is correct. The body is in static equilibrium, and the actual friction is .
   • If : The body is in limiting equilibrium (impending motion).
   • If : The assumption is wrong. The body is in motion. The actual friction acting is kinetic friction, , and dynamic equations () must be used.