Additional: Types of Forces

Weight and Gravitational Force

One of the most universally experienced forces in our daily lives is the force of gravity. This force is responsible for keeping us grounded, making objects fall downwards, and holding celestial bodies in orbit.

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Gravitational Force

The gravitational force is a fundamental non-contact force of attraction that exists between any two objects with mass. According to Newton's Law of Universal Gravitation, the magnitude of the gravitational force ($ F_g $) between two point masses $ m_1 $ and $ m_2 $ separated by a distance $ r $ is given by:

$ F_g = G \frac{m_1 m_2}{r^2} $

where $ G $ is the Universal Gravitational Constant, approximately $ 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $.

This force acts along the line joining the centres of the two masses.

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Weight ($ \vec{W} $)

On the surface of a large celestial body like the Earth, the gravitational force exerted by the Earth on an object is commonly referred to as the object's weight. If an object of mass $ m $ is near the Earth's surface, the Earth's mass ($ M_E $) is concentrated effectively at its center, and the distance $ r $ is approximately the radius of the Earth ($ R_E $).

The gravitational force on the object due to Earth is:

$ F_g = G \frac{M_E m}{R_E^2} $

We know from Newton's Second Law ($ F = ma $) that this gravitational force causes the object to accelerate towards the Earth if no other forces are present (free fall). This acceleration is the acceleration due to gravity, denoted by $ \vec{g} $.

So, $ F_g = mg $.

Comparing the two expressions for $ F_g $:

$ mg = G \frac{M_E m}{R_E^2} $

$ g = G \frac{M_E}{R_E^2} $

This equation shows that the acceleration due to gravity $ g $ is determined by the mass and radius of the Earth (and the gravitational constant $ G $). The average value of $ g $ on the Earth's surface is approximately $ 9.8 \, \text{m/s}^2 $.

The weight of an object ($ \vec{W} $) is the gravitational force acting on it, which is equal to $ m\vec{g} $.

$ \vec{W} = m\vec{g} $

Weight is a vector quantity, always directed downwards towards the centre of the Earth. Its magnitude is $ W = mg $. The SI unit of weight is the Newton (N), as it is a force.

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Distinction between Mass and Weight:

• Mass ($ m $): A scalar quantity, it is the amount of matter in an object and is a measure of its inertia. It is an intrinsic property of the object and does not change with location. Measured in kilograms (kg).
• Weight ($ \vec{W} $): A vector quantity, it is the force of gravity acting on the object. It depends on the mass of the object and the local acceleration due to gravity ($ g $). It changes with location (e.g., weight is different on the Moon compared to Earth, and even slightly different at different altitudes on Earth). Measured in Newtons (N).

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Normal Force

When an object is in contact with a surface, the surface exerts a force on the object. This force is a contact force. The component of this contact force that is perpendicular to the surface is called the normal force.

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Definition and Direction

The normal force, denoted by $ \vec{N} $ (or sometimes $ \vec{F}_N $), is the force exerted by a surface on an object in contact with it, acting in a direction perpendicular (normal) to the surface and pushing away from the surface. It arises from the deformation of the surface and the object at the microscopic level – essentially, the atoms in the surface resist being pushed closer together or distorted by the object.

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Nature of Normal Force

The normal force is a passive force. It is not a fundamental force like gravity or electromagnetism; it is a manifestation of electromagnetic forces between atoms at the interface. Its magnitude is not fixed; it adjusts itself to counteract the forces that are pushing the object into the surface, up to the point where the surface or object starts to deform permanently or break. In many mechanics problems, we assume the surfaces are rigid enough that the normal force adjusts precisely to prevent the object from passing through the surface.

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Normal Force is Not Always Equal to Weight

A common misconception is that the normal force is always equal in magnitude to the weight of the object. This is only true in specific situations, such as when an object is resting on a horizontal surface and there are no other vertical forces acting on it.

Consider an object of mass $ m $ on a horizontal table. Forces acting on the object are weight $ \vec{W} $ (downwards) and normal force $ \vec{N} $ (upwards). Since the object is in vertical equilibrium (not accelerating up or down), the net vertical force is zero:

$ \sum F_y = N - W = 0 \implies N = W = mg $

However, if the object is on an inclined plane at an angle $ \theta $ to the horizontal, the normal force is perpendicular to the plane. The weight vector is vertical. When resolving forces along axes parallel and perpendicular to the incline, the normal force balances only the component of weight perpendicular to the incline ($ W_\perp = W \cos\theta $).

$ N - W \cos\theta = 0 \implies N = W \cos\theta = mg \cos\theta $

In this case, $ N < W $ for $ 0^\circ < \theta \le 90^\circ $.

Other situations where $ N \ne W $ include:

• An object on a horizontal surface with an additional vertical force (e.g., someone pushing down on the object, or pulling it upwards).
• An object in an accelerating elevator.
• An object moving in a vertical circle (e.g., at the top or bottom of a loop).

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Identifying the normal force correctly and understanding its direction is crucial when drawing free body diagrams and applying Newton's Laws, especially on inclined planes or in accelerating frames of reference.

Tension in Strings and Ropes

Strings, ropes, cables, and wires are commonly used in mechanics to transmit forces, often over distances or around pulleys. When a string is pulled taut, it develops a force along its length called tension.

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Definition and Direction

Tension ($ \vec{T} $) is the force transmitted through a string, rope, cable, or similar object when it is pulled tight by forces acting from opposite ends. The tension force always acts along the length of the string and pulls away from the object to which the string is attached.

For example, if an object is hanging from a string, the string exerts an upward tension force on the object. If you pull on a rope attached to a block, the rope exerts a tension force on the block in the direction of the pull.

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Ideal String Assumptions

In many introductory mechanics problems, strings and ropes are treated as "ideal". This involves two main assumptions:

1. Massless: The mass of the string is considered negligible compared to the masses of the objects it connects.

Implication: If the string is massless, then according to Newton's Second Law ($ \vec{F}_{net} = m\vec{a} $), the net force on any segment of the string must be zero ($ \sum \vec{F} = (0)\vec{a} = 0 $). This means the tension force is uniform throughout the string, even if the system is accelerating. The tension pulling on one end of a massless string segment is equal in magnitude to the tension pulling on the other end.

2. Inextensible: The string cannot be stretched or compressed. Its length remains constant.

Implication: All parts of the string, and thus the objects connected by it (if the string remains taut), must move with the same magnitude of velocity and acceleration along the direction of the string. This provides kinematic constraints that link the motions of different objects in a system.

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Effect of a Pulley

A pulley is often used to change the direction of the tension force. If the pulley is considered "ideal" (massless and frictionless axle), the tension in the string remains constant as it passes over the pulley. This allows a force applied in one direction (e.g., hanging weight pulling down) to exert an equal tension force in a different direction (e.g., pulling a block horizontally or up an incline).

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Problems involving tension often require carefully drawing FBDs for each object connected by the string and applying Newton's Second Law to each object, relating their motions via the inextensibility assumption.

Spring Force (Hooke's Law $F = -kx$)

Springs are common mechanical elements that store and release energy when deformed (stretched or compressed). They exert a force called the spring force.

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Hooke's Law

For many springs, within their elastic limit, the force they exert is directly proportional to the amount they are stretched or compressed from their equilibrium length. This relationship is described by Hooke's Law, named after Robert Hooke (1635-1703).

The force exerted by an ideal spring ($ \vec{F}_s $) is given by:

$ \vec{F}_s = -k\vec{x} $

Let's define the terms:

• $ \vec{x} $: This is the displacement vector of the end of the spring from its equilibrium position (the position where the spring is neither stretched nor compressed and exerts no force). The magnitude $ x $ is the amount of stretch or compression.
• $ k $: This is the spring constant (also called the force constant). It is a positive constant that represents the stiffness of the spring. A larger value of $ k $ means a stiffer spring, requiring more force to stretch or compress it by a given amount. The SI unit for $ k $ is Newtons per metre (N/m).
• The negative sign: This is very important. It indicates that the spring force is a restoring force. It always acts in the direction opposite to the displacement. If the spring is stretched ($ \vec{x} $ is in one direction), the spring force pulls back towards the equilibrium position. If the spring is compressed ($ \vec{x} $ is in the opposite direction), the spring force pushes back towards the equilibrium position.

In one dimension, if the equilibrium position is at $ x=0 $, and we consider displacement along the x-axis, Hooke's Law can be written as $ F_x = -kx $, where $ x $ is the displacement from the equilibrium position.

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Limitations of Hooke's Law

Hooke's Law is an idealisation. Real springs follow Hooke's Law only approximately, and only within a certain range of deformation called the elastic limit. If a spring is stretched or compressed beyond its elastic limit, it will undergo permanent deformation and will not return to its original length after the force is removed. In such cases, Hooke's Law is no longer valid.

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Potential Energy Stored in a Spring:

Since the spring force is a conservative force (work done by the spring force depends only on the initial and final positions, not the path taken), we can associate a potential energy with it. The elastic potential energy ($ U_s $) stored in a spring stretched or compressed by a distance $ x $ from its equilibrium position is given by:

$ U_s = \frac{1}{2} k x^2 $

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Spring force is important in understanding oscillations (like Simple Harmonic Motion), shock absorbers, balances, and many other mechanical systems.

Air Resistance and Drag Force

When an object moves through a fluid, such as air or water, the fluid exerts a force on the object that opposes its motion. This force is called fluid resistance or drag force. When the fluid is air, this force is specifically called air resistance.

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Nature and Origin

Drag force is a type of friction that occurs when an object moves through a fluid medium. It arises from two main factors:

1. Viscosity: The internal friction within the fluid and the friction between the fluid and the object's surface. 2. Pressure Difference: As the object moves, it pushes the fluid in front of it, creating a region of higher pressure, and leaves a wake behind it, often with lower pressure. The difference in pressure between the front and back of the object results in a net force opposing the motion.

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Factors Affecting Drag Force

The magnitude of the drag force ($ F_D $) is complex and depends on several factors:

• Speed of the object ($ v $): For slow speeds (laminar flow), drag is roughly proportional to $ v $. For higher speeds (turbulent flow), drag is roughly proportional to $ v^2 $.
• Shape and size of the object: Streamlined shapes experience less drag. Larger cross-sectional area perpendicular to the motion generally results in greater drag.
• Properties of the fluid: Density ($ \rho $) and viscosity ($ \eta $) of the fluid. Denser and more viscous fluids exert greater drag.

A common empirical formula for drag force at moderate to high speeds is:

$ F_D = \frac{1}{2} \rho v^2 C_D A $

where:

• $ \rho $: density of the fluid.
• $ v $: speed of the object relative to the fluid.
• $ C_D $: drag coefficient (a dimensionless quantity that depends on the shape of the object).
• $ A $: reference area (often the projected area of the object perpendicular to the direction of motion).

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Air Resistance in Everyday Life

Air resistance is significant in many situations:

• It causes falling objects to slow down from their initial acceleration due to gravity.
• It opposes the motion of vehicles (cars, bicycles, airplanes, trains), requiring engines to do work to overcome it.
• It affects the trajectory of projectiles.

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Terminal Velocity:

When an object falls through a fluid (like air), the gravitational force pulls it downwards, while the drag force (and possibly buoyancy) acts upwards, opposing the motion. As the object's speed increases, the drag force increases. Eventually, the upward drag force becomes equal in magnitude to the downward gravitational force. At this point, the net force on the object becomes zero ($ \vec{F}_{net} = \vec{W} + \vec{F}_D = 0 $). According to Newton's First Law, when the net force is zero, the acceleration is zero. The object then falls at a constant velocity called the terminal velocity.

For a falling object reaching terminal velocity ($ v_t $), the magnitude of drag force equals the weight (neglecting buoyancy):

$ F_D(v_t) = W $

Using the formula for drag ($ F_D \propto v^2 $ for higher speeds):

$ \frac{1}{2} \rho v_t^2 C_D A = mg $

$ v_t = \sqrt{\frac{2mg}{\rho C_D A}} $

This formula shows that more massive or more streamlined objects (smaller $ C_D A $) have higher terminal velocities. Lighter objects with large areas (like a parachute) have lower terminal velocities.

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