Applying Newton's Laws: Equilibrium, Dynamics, Friction, and Circular Motion

Applying Newton's Laws

Using Newton's First Law: Equilibrium

Newton's First Law states that an object remains at rest or in uniform motion unless acted upon by a net external force. This principle is fundamental for analyzing equilibrium situations.

• Equilibrium: An object is in equilibrium if it is at rest or moving with constant velocity in an inertial frame of reference.

• Mathematical Condition: The vector sum of all forces acting on the object must be zero: $\sum \vec{F} = 0$ In component form: $\sum F_x = 0$ $\sum F_y = 0$

Problem-Solving Strategy for Equilibrium Situations

Set Up:

1. Draw a sketch of the physical situation.

2. Draw a free-body diagram for each object in equilibrium.

3. Identify all forces acting on the object (contact and non-contact). Use $w = mg$ for weight if mass is given.

4. Ensure only forces acting on the object are included.

5. Choose and indicate coordinate axes in your diagram.

Execute:

1. Resolve each force into components along the chosen axes.

2. Set the sum of all x-components and y-components of force to zero separately.

3. For multiple objects, repeat the above steps for each and use Newton's third law for interactions.

4. Ensure the number of independent equations matches the number of unknowns; solve for target variables.

Evaluate:

• Check the physical reasonableness and units of your answer.

Using Newton's Second Law: Dynamics of Particles

Newton's Second Law applies to objects experiencing a net force, causing acceleration.

• Dynamics: The study of objects under the influence of forces resulting in acceleration.

• Newton's Second Law: $\sum \vec{F} = m\vec{a}$ In component form: $\sum F_x = ma_x$ $\sum F_y = ma_y$

• Objects are not in equilibrium when $\sum \vec{F} \neq 0$.

Problem-Solving Strategy for Dynamics Situations

Set Up:

1. Draw a sketch showing each moving object and its forces (free-body diagram).

2. Label all forces (e.g., weight $w = mg$).

3. Choose and indicate coordinate axes for each object.

4. Identify any additional equations (e.g., constraints from ropes or pulleys).

Execute:

1. Resolve forces into components along each axis.

2. List all known and unknown quantities; identify target variables.

3. Write Newton's second law for each component and object; include any additional equations.

4. Solve the equations for the target variables.

Evaluate:

• Check the solution for consistency and correct units.

Free-Body Diagrams

Free-body diagrams are essential tools for visualizing forces acting on an object.

• Key Point: Only actual forces (e.g., gravity, normal, friction, tension) should be included.

• Do not include $ma$ in a free-body diagram; $ma$ is not a force but the result of the net force.

• It is acceptable to indicate the direction of acceleration ($\vec{a}$) beside the diagram for reference.

Frictional Forces

Friction is a force that opposes the relative motion or attempted motion between two surfaces in contact.

• Direction: Friction always acts parallel to the surface and opposite to the direction of motion or impending motion.

• Origin: Friction arises from molecular interactions at the contact surfaces.

Kinetic and Static Friction

• Kinetic Friction ($f_k$): Acts when an object slides over a surface. $f_k = \mu_k n$ where $\mu_k$ is the coefficient of kinetic friction and $n$ is the normal force.

• Static Friction ($f_s$): Acts when there is no relative motion. $f_s \leq \mu_s n$ where $\mu_s$ is the coefficient of static friction. $f_s$ adjusts up to its maximum value to prevent motion.

Transition from Static to Kinetic Friction

• As the applied force increases, static friction increases up to its maximum value ($f_{s,\text{max}} = \mu_s n$).

• Once motion begins, kinetic friction takes over and remains approximately constant ($f_k = \mu_k n$).

Graphical Representation

• Static friction increases linearly with applied force until the maximum is reached; then, kinetic friction acts at a lower, constant value.

Table: Approximate Coefficients of Friction

The following table lists typical values for coefficients of static and kinetic friction for various material pairs:

Materials	Coefficient of Static Friction, $\mu_s$	Coefficient of Kinetic Friction, $\mu_k$
Steel on steel	0.74	0.57
Aluminum on steel	0.61	0.47
Copper on steel	0.53	0.36
Brass on steel	0.51	0.44
Zinc on cast iron	0.85	0.21
Copper on cast iron	1.05	0.29
Glass on glass	0.94	0.40
Copper on glass	0.68	0.53
Teflon on Teflon	0.04	0.04
Teflon on steel	0.04	0.04
Rubber on concrete (dry)	1.0	0.80
Rubber on concrete (wet)	0.30	0.25

Example: Windshield Wipers

• The squeak of windshield wipers on dry glass is due to a stick-slip phenomenon: the blade sticks (static friction), then slips (kinetic friction) when the force exceeds the maximum static friction.

• On wet glass, friction is reduced, so the blade moves smoothly without sticking.

Dynamics of Circular Motion

When a particle moves in a circle at constant speed (uniform circular motion), it experiences a net force directed toward the center of the circle (centripetal force).

• Centripetal Force: The net force required to keep an object moving in a circle of radius $R$ at speed $v$ is $F_{\text{net}} = \frac{mv^2}{R}$

• Direction: Both the acceleration and the net force point toward the center of the circle.

What If the String Breaks?

• If the force maintaining circular motion (e.g., tension in a string) is removed, the object moves in a straight line tangent to the circle, following Newton's first law.

Avoid Using "Centrifugal Force"

• In an inertial frame of reference, there is no real outward (centrifugal) force acting on the object.

• Only the real forces (e.g., tension, gravity, friction) should be included in free-body diagrams.

Example: Car on a Banked Curve

• To determine the angle at which a curve should be banked so a car can make the turn without friction, analyze the forces acting on the car (normal force and gravity) and resolve them into components.

• The banking angle $\theta$ is found by setting the horizontal component of the normal force equal to the required centripetal force: $\tan \theta = \frac{v^2}{Rg}$

Additional info: The notes above are expanded with academic context and examples for clarity and completeness, suitable for college-level physics students.