Newton’s Laws of Motion

Introduction

Newton’s laws of motion are fundamental principles that describe the relationship between forces and the motion of objects. These laws are applicable in various situations, including linear motion and motion in two dimensions.

Newton’s First Law: Law of Inertia

An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by a force.

• Inertia: The tendency of an object to resist changes in its state of motion.
• An object will maintain its velocity (both speed and direction) unless a net external force acts on it.
• Inertia is directly proportional to mass. A more massive object has more inertia.

KEY TAKEAWAY: An object’s inertia resists changes to its velocity.

Newton’s Second Law: Law of Acceleration

The acceleration of an object is directly proportional to the net force acting on the object, is in the same direction as the net force, and is inversely proportional to the mass of the object.

• Mathematical representation:

  \[F_{net} = ma\]

  Where:

  • \(F_{net}\) is the net force acting on the object (N)
  • \(m\) is the mass of the object (kg)
  • \(a\) is the acceleration of the object (m/s²)
  • The net force is the vector sum of all forces acting on the object.
  • If \(F_{net} = 0\), then \(a = 0\), which means the object is either at rest or moving with constant velocity (consistent with Newton’s First Law).

EXAM TIP: Remember to use vector addition to find the net force when multiple forces are acting on an object.

Applying Newton’s Second Law

1. Identify all forces acting on the object.
2. Draw a free-body diagram representing the forces as vectors.
3. Choose a coordinate system (x and y axes).
4. Resolve forces into components along the chosen axes.

5. Apply Newton’s Second Law separately for each axis:

   • \(\sum F_x = ma_x\)
   • \(\sum F_y = ma_y\)
   • Solve for the unknowns, such as acceleration or force components.

COMMON MISTAKE: Forgetting to consider all forces acting on the object. Always draw a free-body diagram!

Newton’s Third Law: Law of Action-Reaction

For every action, there is an equal and opposite reaction.

• When object A exerts a force on object B, object B simultaneously exerts an equal and opposite force on object A.
• These forces act on different objects.
• Action-reaction pairs never act on the same object. If they did, the net force would always be zero, and no acceleration would ever occur.
• Examples:
  • A person walking pushes on the ground (action), and the ground pushes back on the person (reaction).
  • A rocket expels exhaust gases (action), and the gases exert a force on the rocket, propelling it forward (reaction).

STUDY HINT: Practice identifying action-reaction pairs in different scenarios.

Coplanar Forces

Coplanar forces are forces that act on an object in the same plane (two-dimensional).

Resultant Force

The resultant force (or net force) is the vector sum of all coplanar forces acting on an object. To find the resultant force:

1. Resolve each force into its x and y components.
2. Sum the x-components to find the x-component of the resultant force (\(R_x\)).
3. Sum the y-components to find the y-component of the resultant force (\(R_y\)).

4. Calculate the magnitude of the resultant force using the Pythagorean theorem:

   \$\(R = \sqrt{R_x^2 + R_y^2}\)\$
   5. Determine the direction of the resultant force using trigonometry:

   \[ \theta = \tan^{-1} \left( \frac{R_y}{R_x} \right) \]

   Where \(\theta\) is the angle the resultant force makes with the x-axis.

APPLICATION: Calculating the net force on a car being pulled by multiple ropes or on an object suspended by cables.

Equilibrium

An object is in equilibrium when the net force acting on it is zero. This means that the object is either at rest (static equilibrium) or moving with constant velocity (dynamic equilibrium).

• For an object in equilibrium:

  \[\sum F_x = 0\]
  \[\sum F_y = 0\]

Applications in One and Two Dimensions

One-Dimensional Motion

• Forces act along a single line.
• Example: A block being pulled along a horizontal surface with friction.

Two-Dimensional Motion

• Forces act in a plane, requiring vector resolution.
• Examples:
  • An object on an inclined plane: gravity, normal force, and friction.
  • An object suspended by two ropes: tension in each rope and gravity.
  • Projectile motion (covered in more detail elsewhere, but Newton’s laws are the foundation).

Inclined Planes

• Forces involved: Gravity (\(F_g\)), Normal Force (\(F_N\)), Friction (\(F_f\))
• Coordinate System: Choose a coordinate system where the x-axis is parallel to the inclined plane and the y-axis is perpendicular to the plane.

• Resolve Gravity: Resolve the force of gravity into components parallel (\(F_{g||}\)) and perpendicular (\(F_{g\perp}\)) to the plane:

  • \(F_{g||} = mg\sin\theta\)
  • \(F_{g\perp} = mg\cos\theta\)

  Where \(\theta\) is the angle of the incline.
  * Apply Newton’s Second Law:

  • \(\sum F_x = F_{g||} - F_f = ma_x\)
  • \(\sum F_y = F_N - F_{g\perp} = 0\) (since there is no acceleration in the y-direction)

Connected Bodies

• When two or more objects are connected (e.g., by a rope), they often have the same magnitude of acceleration.
• Steps to analyze:
  1. Draw free-body diagrams for each object separately.
  2. Identify the forces acting on each object.
  3. Apply Newton’s Second Law to each object.
  4. Use the fact that the acceleration is the same (in magnitude) for all connected objects.
  5. Solve the system of equations to find the unknowns (e.g., acceleration, tension in the rope).

VCAA FOCUS: Be prepared to analyze scenarios involving inclined planes and connected bodies on exams.