Unit 2: Force and Translational Dynamics

Inertial Mass vs. Gravitational Mass

• Inertial Mass: The measure of an object's inertia or resistance to acceleration.
• Gravitational Mass: Used to determine the force of gravity or weight of an object.
  • $F_g = mg$
• Inertial Mass and Gravitational Mass are experimentally identical.

Newton's Laws of Motion

Newton’s First Law:

“An object at rest will remain at rest and an object in motion will remain at a constant velocity unless acted upon by an external force.”

• Common mistake: “An object in motion will remain in motion” is incorrect. It will remain at a constant velocity, meaning it will have a constant speed and a constant direction.
• Common mistake: "Unless acted upon by an external force." The key term is net force. It is the sum of all the forces that needs to be zero for an object to remain at rest or at constant velocity.

Newton’s Second Law:

$$\sum \mathbf{F} = m\mathbf{a}$$

• It is also written as:

  $$\mathbf{a} = \frac{\sum \mathbf{F}}{m}$$

• When using Newton’s Second Law, you must identify object(s) and direction.
• Free Body Diagrams: Always draw them to use Newton’s Second Law.
  • On the AP Test, do NOT break forces into components in your initial Free Body Diagram.

Forces

• The Force of Gravity (or Weight) of an object is always downward:

  $$F_g = mg$$

• The Normal Force is caused by a surface, is always normal (perpendicular) to the surface, and is always a push.

Units of Force

Force is measured in Newtons (N):

$$\sum \mathbf{F} = m\mathbf{a} \Rightarrow N = kg \cdot m/s^2$$

Force of Friction

• Friction is parallel to the surface, opposes motion, and is independent of the direction of the applied force.
• The equation for friction is:

  $$|F_f| \leq \mu |F_n|$$

• There are two types of friction:
  • Static (non-moving) friction: The two surfaces do not slide relative to one another.

    $$F_s \leq \mu_s F_n \quad \text{and} \quad F_{s,\text{max}} = \mu_s F_n$$

  • Kinetic (moving) friction: The two surfaces do slide relative to one another.

    $$F_k = \mu_k F_n$$

• The coefficient of kinetic friction is always less than that of static friction:

  $$\mu_k < \mu_s$$

Newton’s Third Law:

$$F_{A \to B} = -F_{B \to A}$$

For every force from object A on object B, there is an equal but opposite force from object B on object A.

Newton’s Third Law Force Pairs (Action-Reaction Pairs):

• They act on two different objects simultaneously.

Inclines

• The force of gravity in its components that are parallel and perpendicular to an incline:

  $$F_g = mg \sin\theta, \quad F_n = mg \cos\theta$$

Translational Equilibrium

$$\sum \mathbf{F} = 0 \Rightarrow m\mathbf{a} = 0$$

This means the object is either at rest or moving with a constant velocity.