Lecture 3: Relative Velocity, Newton's Laws, Friction, and Free Body Diagrams

Relative Velocity

Understanding Relative Velocity

Relative velocity describes the velocity of an object as observed from a particular reference frame. It is crucial in analyzing motion in scenarios involving moving media, such as swimmers in a river or airplanes in the wind.

• Key Terms:

  • Reference Frame: The perspective from which motion is observed and measured.

  • Resultant Velocity: The net velocity of an object, accounting for all contributing velocities (e.g., swimmer's speed plus water current).

• Vector Addition: Relative velocities are combined using vector addition.

Equation:

$\vec{v}_{SW} + \vec{v}_{WL} = \vec{v}_{SL}$

• $\vec{v}_{SW}$: Velocity of swimmer relative to water

• $\vec{v}_{WL}$: Velocity of water relative to land

• $\vec{v}_{SL}$: Velocity of swimmer relative to land

Example: Ferryboat Crossing

• A ferry crosses a 1 km channel in 10 minutes on a calm day ($6\ \text{km/h}$). With a $4\ \text{km/h}$ current, the resultant velocity is found using vector addition:

$\vec{v}_{FG}^2 = 4^2 + 6^2 = 4.47\ \text{km/h}$ $t = \frac{1\ \text{km}}{4.47\ \text{km/h}} = 0.22\ \text{h} = 13\ \text{minutes}$

• Application: This method applies to swimmers, boats, and airplanes affected by wind or current.

Comments on Relative Velocity

• One velocity is often the 'wind speed' or 'water speed' (not the resultant).

• Another is the object's speed in still conditions.

• The resultant is the combination, determining the actual path.

Practice Problems

• Paper airplane problems illustrate vector addition when air currents affect the path.

Force and Motion

What is a Force?

A force is an interaction between two bodies that can cause acceleration. It is a vector quantity, meaning it has both magnitude and direction. Forces can be produced with or without contact.

• SI Unit: Newton (N), where $1\ \text{N} = 1\ \text{kg} \cdot \text{m}/\text{s}^2$

Fundamental Forces

• Gravity

• Electromagnetic

• Strong Nuclear

• Weak Nuclear

In introductory physics, focus is on gravity and electromagnetic forces (manifested as normal, tension, and friction forces).

• Contact Forces: Normal, tension, and friction require contact.

• Action-at-a-Distance: Gravity, magnetic, and electrostatic forces can act without contact.

Newton's Laws of Motion

Newton's First Law (Law of Inertia)

If the net external force on an object is zero, the object remains at rest or moves with constant velocity in a straight line.

$\frac{\Delta \vec{v}}{\Delta t} = \vec{a} = 0$ $\vec{F}_{\text{net}} = m\vec{a} = 0$

• Used to analyze equilibrium situations (no acceleration).

Newton's Second Law

A net force acting on an object of mass $m$ causes an acceleration $a$ in the direction of the force.

$\vec{F} = m\vec{a}$ $\vec{F}_{\text{net}} = \sum \vec{F}_i = m\vec{a}$

• Used when acceleration is present (non-equilibrium).

Newton's Third Law

If two objects interact, the force exerted by object A on object B is equal in magnitude and opposite in direction to the force exerted by object B on object A.

$\vec{F}_{A\ on\ B} = -\vec{F}_{B\ on\ A}$

• Action-reaction pairs act on different objects.

Examples of Action-Reaction Pairs

• Block pushes down on table; table pushes up on block.

• You push on a block; block pushes back on you.

Gravitational Force

Any two objects with mass exert an attractive force on each other, proportional to their masses and inversely proportional to the square of their separation.

$|\vec{F}_g| = G \frac{m_1 m_2}{r^2}$ $g = \frac{G m_e}{r_e^2} \approx 9.8\ \text{m/s}^2$ $|\vec{F}_g| = m_1 g = |\vec{w}|$

• Application: Weight is the gravitational force near Earth's surface.

Friction

Static Friction ($\vec{f}_s$)

Occurs when an object is not moving. It points opposite to the intended motion and adjusts up to a maximum value to prevent motion.

$f_s \leq \mu_s N$

• $\mu_s$ is the coefficient of static friction, $N$ is the normal force.

• Static friction ensures $\sum F = 0$ for objects at rest.

Kinetic Friction ($\vec{f}_k$)

Occurs when an object is moving. The magnitude is constant and given by:

$|\vec{f}_k| = \mu_k N$

• $\mu_k$ is the coefficient of kinetic friction.

• Kinetic friction opposes the direction of motion.

Friction Force vs. Applied Force

Region	Friction Force	Description
Static	$f_s \leq \mu_s N$	Object at rest; friction increases with applied force up to maximum
Kinetic	$f_k = \mu_k N$	Object in motion; friction is constant

Free Body Diagrams (FBDs)

Drawing Free Body Diagrams

A Free Body Diagram represents all forces acting on a single object. It is a crucial tool for solving force problems.

• Draw only forces acting on the object of interest.

• Do not include forces the object exerts on others, internal forces, or a force for the $ma$ term.

• Draw arrows pointing outward from the object.

Solving Force Problems

1. Draw a Free Body Diagram (FBD).

2. Define x and y axes.

3. Apply Newton's 1st or 2nd Law in x and y directions.

4. Solve the resulting equations for unknowns.

$\sum \vec{F} = 0$ (no acceleration, constant velocity) $\sum \vec{F} = m\vec{a}$ (acceleration present)

Practice Examples

• Inclined Plane (Frictionless): Block must accelerate down the incline.

• Inclined Plane (With Friction): Friction can oppose motion or keep the block at rest.

• Multiple Forces: Identify all forces acting on an object, such as normal, friction, gravity, and applied forces.

Worked Examples

• Spider Hanging from a Thread: Find tension and wind force using equilibrium equations.

• Mass Suspended by Two Strings: Use equilibrium in x and y directions to solve for tensions.

• Elevator Problem: Calculate apparent weight under different acceleration scenarios.

Table: Apparent Weight in an Elevator

Scenario	Apparent Weight Equation
Ascending, constant velocity	$N = mg$
Descending, constant velocity	$N = mg$
Ascending, acceleration $a$	$N = m(g + a)$
Descending, acceleration $a$	$N = m(g - a)$

Additional info: The notes also include clicker quizzes and practice problems to reinforce concepts, as well as diagrams illustrating force vectors and free body diagrams.