Chapter 4: The Laws of Motion

4.1 Forces

Forces are interactions that can change the motion of objects. There are four known fundamental forces in nature: strong nuclear, electromagnetic, weak nuclear, and gravitational forces. In classical physics, we primarily focus on gravitational and electromagnetic forces.

• Force: A push or pull acting on an object, measured in newtons (N).

• Contact forces arise from physical contact (e.g., friction, tension).

• Field forces act at a distance (e.g., gravity, electric force).

Newton's universal law of gravitation describes the gravitational force between two masses:

where is the gravitational constant, and are masses, and is the distance between their centers.

4.2 Newton's First Law

Newton's First Law (Law of Inertia) states that an object at rest remains at rest, and an object in motion continues in motion with constant velocity unless acted upon by a net external force.

• Inertia: The tendency of an object to resist changes in its state of motion.

• Mass: A measure of an object's inertia; SI unit is the kilogram (kg).

4.3 Newton's Second Law

Newton's Second Law quantifies the relationship between force, mass, and acceleration:

• The net force acting on an object equals the product of its mass and acceleration .

• Acceleration is in the direction of the net force.

4.4 Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction:

• If object A exerts a force on object B, then B exerts an equal and opposite force on A.

• Action and reaction forces act on different objects.

4.5 Applications of Newton's Laws

Newton's laws are used to analyze equilibrium and non-equilibrium situations:

• Equilibrium: The net force on an object is zero (), so the object is at rest or moves with constant velocity.

• Non-equilibrium: The net force is not zero, resulting in acceleration ().

• Free-body diagrams are essential tools for visualizing forces acting on objects.

4.6 Forces of Friction

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

• Static friction (): Prevents motion up to a maximum value.

• Kinetic friction (): Acts when objects are sliding past each other.

The maximum static friction force is:

The kinetic friction force is:

where and are the coefficients of static and kinetic friction, and is the normal force.

Chapter 5: Energy

5.1 Work

Work is done when a force causes displacement in the direction of the force:

• is the magnitude of the force, is the angle between force and displacement, is the displacement.

• Work is measured in joules (J).

5.2 Kinetic Energy and the Work-Energy Theorem

Kinetic energy () is the energy of motion:

The work-energy theorem states:

• The net work done on an object equals the change in its kinetic energy.

5.3 Gravitational Potential Energy

Potential energy is stored energy due to position. Gravitational potential energy () is:

• is mass, is acceleration due to gravity, is height above a reference point.

5.4 Spring Potential Energy

Elastic potential energy is stored in a stretched or compressed spring:

• is the spring constant, is the displacement from equilibrium.

5.5 Systems and Energy Conservation

The principle of conservation of energy states that the total energy of an isolated system remains constant:

• Where is kinetic energy, is potential energy, and is work done by non-conservative forces (e.g., friction).

5.6 Power

Power is the rate at which work is done or energy is transferred:

• Measured in watts (W), where 1 W = 1 J/s.

Chapter 6: Momentum and Collisions

6.1 Momentum and Impulse

Momentum () is the product of mass and velocity:

Impulse () is the change in momentum caused by a force acting over a time interval:

6.2 Conservation of Momentum

In an isolated system (no external forces), total momentum is conserved:

• Applies to all types of collisions.

6.3 Collisions

Collisions are classified as elastic or inelastic:

• Elastic collision: Both momentum and kinetic energy are conserved.

• Inelastic collision: Only momentum is conserved; kinetic energy is not.

For a one-dimensional elastic collision between two objects:

6.4 Glancing Collisions

In two-dimensional collisions, momentum is conserved in both the x and y directions. The final velocities can be found using vector components and trigonometry.

Chapter 7: Rotational Motion

7.1 Angular Speed and Angular Acceleration

Rotational motion describes the movement of objects around a fixed axis.

• Average angular speed ():

• Average angular acceleration ():

7.2 Rotational Motion Under Constant Angular Acceleration

For constant angular acceleration, the following equations describe rotational kinematics:

These equations are analogous to linear kinematics equations.

Table: Comparison of Force Types

Example Problems

• Example 1: A freight train of mass kg is pulled by a constant force of N. Find the time to accelerate from rest to 80 km/h.

  • Use to find acceleration, then to solve for time.

• Example 2: A 75-kg man in an elevator reads 825 N on a scale as the elevator rises. Find the acceleration.

  • Apply and solve for .

• Example 3: A 0.50-kg ball falls from 30 m and rebounds to 20 m. If contact with the ground lasts 2.0 ms, find the average force exerted.

  • Use impulse-momentum theorem: .

• Example 4: A 150-N bird feeder is supported by three cables. Find the tension in each cable.

  • Draw a free-body diagram and apply equilibrium conditions (, ).

• Example 5: Drops of rain (0.035 kg/s) strike a roof at 12 m/s and come to rest. Find the average force exerted by the rain.

  • Use .

Additional info: The above examples are representative of typical problems found in introductory physics courses, illustrating the application of Newton's laws, work-energy principles, and momentum conservation.