Dynamics

Dynamics is the branch of mechanics that deals with the study of forces and their effects on motion. While kinematics describes the motion of objects without considering the causes, dynamics focuses on understanding why objects move the way they do.

The foundation of dynamics lies in Newton's laws of motion, which provide a framework for relating forces to the resulting motion of objects.

Force is a push or pull that can cause an object to accelerate. It is a vector quantity, having both magnitude and direction. The SI unit of force is the newton (N), which is defined as the force needed to accelerate a 1 kg mass at a rate of 1 m/s².

Mass is a measure of an object's inertia, or resistance to acceleration when a force is applied. The SI unit of mass is the kilogram (kg).

It's important to distinguish between mass and weight:

• Mass is an intrinsic property of matter and remains constant regardless of location
• Weight is the force of gravity acting on an object and varies depending on the gravitational field strength

The relationship between weight (W) and mass (m) is given by:

W = mg

Where g is the gravitational field strength (approximately 9.8 N/kg on Earth's surface).

A free body diagram (FBD) is a simplified representation of an object showing all the external forces acting on it. Creating an FBD is often the first step in solving dynamics problems.

Steps to create a free body diagram:

1. Isolate the object of interest
2. Identify all external forces acting on the object
3. Represent each force as a vector arrow pointing in the direction of the force
4. Label each force with its magnitude and direction

Common forces to include in FBDs:

• Weight (gravitational force)
• Normal force (perpendicular to surfaces in contact)
• Friction (parallel to surfaces in contact)
• Tension (in strings, ropes, or cables)
• Applied forces (pushes or pulls)
• Spring forces

Newton's Second Law (F = ma) is the fundamental equation used in dynamics to relate forces to motion. When applying this law:

1. Draw a free body diagram
2. Choose a coordinate system (typically with one axis in the direction of acceleration)
3. Resolve all forces into components along your coordinate axes
4. Apply F = ma along each axis
5. Solve the resulting equations

For multiple objects connected together (e.g., by ropes or springs), you need to:

• Draw separate FBDs for each object
• Apply F = ma to each object individually
• Identify constraint equations that relate the motions of different objects

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

Static friction acts between surfaces that are not moving relative to each other. It prevents objects from starting to slide. The maximum static friction is given by:

f_s,max = μ_sN

Kinetic friction acts between surfaces that are sliding relative to each other. It is given by:

f_k = μ_kN

Where:

• μ_s is the coefficient of static friction
• μ_k is the coefficient of kinetic friction
• N is the normal force

Generally, μ_s > μ_k, meaning it takes more force to start an object sliding than to keep it sliding.

When an object moves in a circular path, it experiences an acceleration toward the center of the circle called centripetal acceleration:

a_c = v²/r

According to Newton's Second Law, a force must cause this acceleration. This force, directed toward the center of the circle, is called the centripetal force:

F_c = ma_c = mv²/r

The centripetal force is not a new type of force; it's provided by existing forces such as:

• Tension in a string (for a ball swung in a circle)
• Gravitational force (for planets orbiting the Sun)
• Friction or normal force (for a car turning on a road)

For a car turning on a flat road, the centripetal force is provided by friction between the tires and the road. For a car turning on a banked curve, both friction and the normal force contribute to the centripetal force.