Problem 48 - Master Resource Book in JEE Main Physics

Question

The maximum static friction on a body is \(F=\mu N\). Here, \(N=\) normal reaction force on the body \(\mu=\) coefficient of static friction. The dimensions of \(\mu\) are (a) \(\left[\mathrm{MLT}^{-2}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{~L}^{\circ} \mathrm{T}^{0} \theta^{-1}\right]\) [c) dimensionless (d) None of these

Step-by-Step Solution

Answer

The coefficient of static friction \(\mu\) is dimensionless.

Step 1: Understanding the Problem

We need to find the dimensions of the coefficient of static friction \(\mu\) when given that the maximum static friction \(F\) is expressed as \(F=\mu N\), where \(N\) is the normal reaction force.

Step 2: Analyzing the Equation

From \(F=\mu N\), \(\mu\) can be expressed as \(\mu = \frac{F}{N}\). Therefore, we need to compare the dimensions of force \(F\) and normal force \(N\) in order to find the dimensions of \(\mu\).

Step 3: Determining Dimensions of Force

The dimensions of force \(F\) are given by \(\left[ \text{MLT}^{-2} \right]\). This is based on Newton's second law (\(F=ma\)).

Step 4: Determining Dimensions of Normal Reaction Force

The normal reaction force \(N\) also has the dimensions of \(\left[ \text{MLT}^{-2} \right]\), as it is also a force and usually corresponds to \(mg\) for an object resting on a surface, where \(m\) is mass and \(g\) is acceleration due to gravity.

Step 5: Calculating the Dimensions of \(\mu\)

Since \(\mu = \frac{F}{N}\), substituting the dimensions we get \(\mu = \frac{\left[ \text{MLT}^{-2} \right]}{\left[ \text{MLT}^{-2} \right]}\), which simplifies to \(\left[ \text{M}^0 \text{L}^0 \text{T}^0 \right]\), showing that \(\mu\) is dimensionless.

Key Concepts

Static Friction

Static friction is an important concept in understanding how objects resist the start of motion. This frictional force acts between two surfaces when an attempt is made to move one of them over the other without causing actual movement. - Static friction is generally higher than kinetic friction, which is the friction present when objects are sliding past each other.- It acts to prevent motion up to a certain applied force limit; beyond this point, motion occurs as kinetic friction takes over. - Static friction can be mathematically expressed using the formula: \[ F_{s} \leq \mu_{s} N \] Where: - \( F_{s} \) is the static friction force - \( \mu_{s} \) is the coefficient of static friction - \( N \) is the normal forceThe coefficient of static friction \( \mu_{s} \) is a measure of the adhere-ability and physical texture of the surfaces in contact before they move relative to each other. A higher \( \mu_{s} \) indicates more resistance to sliding motion. Thus, understanding static friction is crucial in applications like determining the stopping distances of vehicles and analyzing objects' stability under various conditions.

Normal Force

The normal force is a support force exerted by a surface perpendicular to an object resting on it. It's crucial in balancing forces perpendicular to the surface and plays a fundamental role in calculating friction.- When an object is placed on a horizontal surface, the normal force is equal in magnitude and opposite in direction to its weight (gravitational force).- In inclined planes, the normal force is less than the object's weight and acts parallel to the inclined surface.Normal force (\( N \)) can often be calculated as:\[ N = mg \cos \theta \]Where:- \( m \) is the mass of the object- \( g \) is the acceleration due to gravity- \( \theta \) is the angle of the incline, if applicableThe significance of the normal force goes beyond just opposing gravity. It enables us to compute frictional forces — both static and kinetic — accurately, making it indispensable in problems involving motion analysis.

Dimensionless Quantity

A dimensionless quantity is a type of physical quantity that has no physical dimensions. It is expressed as a simple number without any unit. This concept is crucial in various fields of science and engineering, providing a basis for comparing magnitudes or rates without the complexity of units.- Examples of dimensionless quantities include coefficients of friction, angles measured in radians, and refractive indices in optics.A dimensionless quantity arises when you divide one dimensional quantity by another of the same dimensional type, thus cancelling out their dimensions.For instance, in the formula for maximum static friction, \( F = \mu N \), the coefficient of static friction \( \mu \) is dimensionless because:\[ \mu = \frac{F}{N} = \frac{[\text{MLT}^{-2}]}{[\text{MLT}^{-2}]} = [\text{M}^0 \text{L}^0 \text{T}^0] \]This shows \( \mu \) is merely a ratio of forces with no inherent scale or magnitude on its own. Dimensionless quantities are instrumental in constructing dimensionless numbers and scaling analyses, both of which hold significant importance in theoretical and applied physics.