Newton's Laws of Motion

2.1 Force and Interactions

Forces are interactions between two objects. A force is a vector quantity, meaning it has both magnitude and direction. Forces can be classified as contact forces (e.g., push, pull, friction, tension) or long-range forces (e.g., gravitational, magnetic).

• Contact Forces: Require physical contact between objects (e.g., friction, tension in a rope).

• Long-range Forces: Act at a distance without direct contact (e.g., gravity, magnetism).

• Unit of Force: The SI unit is the newton (N).

Example: Tension in a rope pulling an object, or the gravitational force between the Earth and an object.

2.2 Combining Forces (Vector Addition)

When multiple forces act on an object, the net effect is the vector sum of all individual forces. This is called the resultant force.

• The net force is given by:

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• Forces can be resolved into components, typically along the x and y axes:

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• The magnitude of the resultant force is:

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• The direction is given by:

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Example: Calculating the resultant of three forces acting at different angles using vector components.

2.3 Newton's First Law (Law of Inertia)

Statement: An object acted on by no net external force has a constant velocity (which may be zero); it does not accelerate.

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• This means that if all forces cancel out, the object remains at rest or moves with constant velocity.

• Inertia is the property of an object to resist changes in its state of motion.

Example: A passenger in a car continues moving forward when the car suddenly stops due to inertia.

2.4 Newton's Second Law

Statement: If a net external force acts on an object, the object accelerates in the direction of the net force. The acceleration is proportional to the net force and inversely proportional to the object's mass.

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• This is a vector equation and can be written in component form:

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• The law applies only in inertial frames of reference (non-accelerating frames).

Example: Calculating the mass of a block of ice given the net force and acceleration, or determining the acceleration of a system when a force is applied.

2.5 Mass and Weight

Mass is a measure of an object's inertia—its resistance to acceleration. Weight is the gravitational force exerted on an object by the Earth.

• Weight is given by:

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• Where is the acceleration due to gravity ( on Earth).

Example: Calculating the weight of a 10 kg object: .

2.6 Newton's Third Law

Statement: For every action, there is an equal and opposite reaction. If object A exerts a force on object B, then object B exerts a force of equal magnitude and opposite direction on object A.

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• Action and reaction forces act on different objects.

Example: When you push against a wall, the wall pushes back with an equal and opposite force.

2.7 Equilibrium and Newton's First Law

When the net force on an object is zero, the object is in equilibrium. This means it is either at rest or moving with constant velocity.

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• For equilibrium in two dimensions:

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Example: Calculating the tension in cables supporting a stationary wrecking ball.

2.8 Dynamics of Particles (Newton's Second Law in Non-Equilibrium)

When the net external force is not zero, the object accelerates according to Newton's Second Law.

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Example: Calculating the acceleration and tension in a cable for a person standing on a scale in an accelerating elevator.

Friction Forces

2.9.1 Kinetic and Static Friction

Friction is a force that opposes the relative motion of two surfaces in contact. There are two main types:

• Static Friction (): Acts when an object is not moving relative to the surface. It prevents motion up to a maximum value.

• Kinetic Friction (): Acts when an object is sliding over a surface. It usually has a constant value less than the maximum static friction.

• and are the coefficients of static and kinetic friction, respectively.

• is the normal force (perpendicular to the surface).

Example: Calculating the force required to start moving a crate and the force needed to keep it moving at constant speed.

2.9.2 Fluid Resistance and Terminal Speed

Fluid resistance (drag) is a force exerted by a fluid on an object moving through it. For small objects at low speeds, the force is proportional to velocity:

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• At higher speeds, the force is proportional to :

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• Terminal speed is reached when the drag force equals the gravitational force:

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Example: Calculating the terminal speed of a falling object in air.

Dynamics of Circular Motion

2.10 Dynamics of Circular Motion

For an object moving in a circle of radius with constant speed , the net force required to keep it in circular motion is called the centripetal force:

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• This force is always directed toward the center of the circle.

Example: Tension in a string for a conical pendulum, or the normal force on a car rounding a banked curve.

Example Table: Friction Forces

Key Example Problems

• Finding the resultant of several forces acting at angles using vector components.

• Calculating the acceleration and tension in a two-mass system connected by a string over a pulley.

• Determining the force required to move a crate, the coefficients of friction, and the acceleration when a push is applied.

• Analyzing the forces on a passenger in a car rounding a curve or moving in a vertical circle (e.g., Ferris wheel).

Additional info: The notes include worked examples with step-by-step solutions, reinforcing the application of Newton's laws to real-world problems involving forces, friction, and circular motion.