Physics Study Guide: Newton's Laws, Forces, Gravitation, Work, Energy, and Power

Q1. What are Newton’s three laws of motion? Express all three laws in words.

Background

Topic: Newton's Laws of Motion

This question tests your understanding of the foundational principles governing classical mechanics. Newton's three laws describe how objects move and interact with forces.

Key Terms and Concepts:

• Newton's First Law (Law of Inertia)

• Newton's Second Law (Law of Acceleration)

• Newton's Third Law (Action-Reaction Law)

Step-by-Step Guidance

1. State Newton's First Law in your own words, focusing on the concept of inertia and what happens when no net force acts on an object.

2. Express Newton's Second Law in words, emphasizing the relationship between force, mass, and acceleration.

3. Describe Newton's Third Law, highlighting the idea of action and reaction forces between two objects.

Try writing out each law in your own words before checking the answer!

Q1a. Express Newton’s second law mathematically. What does this equation look like as a general vector relationship? What does the equation look like as a one-dimensional vector relationship? What does the equation look like as a relationship between vector magnitudes?

Background

Topic: Newton's Second Law (Mathematical Forms)

This question asks you to translate Newton's second law from words into mathematical equations, considering both vector and scalar forms.

Key Formulas:

• General vector form: $\vec{F}_{\text{net}} = m \vec{a}$

• One-dimensional vector form: $F_{\text{net},x} = m a_x$

• Magnitude (scalar) form: $F_{\text{net}} = m a$

Step-by-Step Guidance

1. Write the general vector equation for Newton's second law, using vector notation for force and acceleration.

2. Adapt the equation for motion along a single axis (e.g., x-direction), showing how the vector equation simplifies.

3. Express the equation in terms of magnitudes, assuming force and acceleration are in the same direction.

Try expressing each form before revealing the answer!

Q1b. Use real-world examples to describe Newton’s first and third laws.

Background

Topic: Application of Newton's Laws

This question asks you to connect the abstract laws to everyday experiences, demonstrating your conceptual understanding.

Key Concepts:

• Inertia (First Law)

• Action-Reaction Pairs (Third Law)

Step-by-Step Guidance

1. Think of a situation where an object remains at rest or in motion unless acted on by a force (First Law).

2. Consider an example where two objects exert equal and opposite forces on each other (Third Law).

3. Describe each example clearly, identifying the forces involved.

Try coming up with your own examples before checking the answer!

Q2. Define weight; define mass. How are mass and weight related; how are they different?

Background

Topic: Mass vs. Weight

This question tests your understanding of the distinction between mass (a measure of matter) and weight (the force due to gravity).

Key Terms and Formulas:

• Mass ($m$): Scalar quantity, measured in kilograms (kg).

• Weight ($W$): Force due to gravity, measured in newtons (N).

• Relationship: $W = m g$

Step-by-Step Guidance

1. Define mass in your own words, including its units and whether it changes with location.

2. Define weight, including its units and dependence on gravitational acceleration.

3. Write the formula relating weight and mass, and explain what each symbol represents.

4. Discuss how mass and weight differ conceptually and practically.

Try explaining the differences before revealing the answer!

Q2. For each of the following “forces with names” describe each force and when each must be considered in a “sum of forces” statement of Newton’s second law. What is the magnitude of the force (is it consistent, or does it depend on the circumstances)? What is consistent about the direction of each force?

Background

Topic: Types of Forces in Mechanics

This question asks you to describe common forces (gravity, normal, tension, friction), when to include them in force analysis, and their properties.

Key Terms:

• Gravity: $F_g = m g$

• Normal Force

• Tension

• Friction: $f = \mu N$ (for kinetic friction)

Step-by-Step Guidance

1. For each force, describe what it is and when it acts on an object.

2. State whether the magnitude is constant or depends on the situation (e.g., friction depends on normal force).

3. Describe the direction of each force and what determines it.

4. Explain when each force should be included in the sum of forces for Newton's second law.

Try describing each force before checking the answer!

Q3. How do we use a free body diagram to assist in problem solving? What should we consider in our choice of a frame of reference? How do we use Newton’s second law to form one-dimensional vector relationships?

Background

Topic: Free Body Diagrams and Problem Solving

This question tests your ability to use diagrams and coordinate systems to analyze forces and motion.

Key Concepts:

• Free Body Diagram (FBD)

• Frame of Reference

• Newton's Second Law in 1D: $\sum F_x = m a_x$

Step-by-Step Guidance

1. Describe the steps to draw a free body diagram for an object.

2. Explain how to choose a frame of reference (coordinate axes) for the problem.

3. Show how to write Newton's second law as a sum of forces along one axis.

Try drawing an FBD and setting up the equation before revealing the answer!

Q1. Understand the magnitude of the force of gravity on an object on planet earth in the context of “Universal Gravitation”: Why is the force of gravity on earth the way it is?

Background

Topic: Universal Gravitation

This question explores why the gravitational force on Earth has its observed value, connecting Newton's law of universal gravitation to everyday experience.

Key Formula:

• Newton's Law of Universal Gravitation: $F = G \frac{m_1 m_2}{r^2}$

• On Earth's surface: $F = m g$

Step-by-Step Guidance

1. Write the universal gravitation formula and identify each variable.

2. Explain how this formula applies to an object near Earth's surface (using Earth's mass and radius).

3. Show how the formula simplifies to $F = m g$ for objects on Earth, and discuss why $g$ has its value.

Try connecting the universal law to Earth's gravity before revealing the answer!

Q2. Describe “Universal Gravitation”: What is the direction of the force of gravity between any two objects having mass? What is the magnitude of the force of gravity between any two massive objects?

Background

Topic: Newton's Law of Universal Gravitation

This question asks you to explain the direction and magnitude of gravitational force between masses.

Key Formula:

• $F = G \frac{m_1 m_2}{r^2}$

Step-by-Step Guidance

1. State the direction of the gravitational force between two masses (always attractive, along the line joining centers).

2. Write the formula for the magnitude of the gravitational force and define each variable.

3. Discuss how the force changes with mass and distance.

Try explaining both direction and magnitude before revealing the answer!

Q1. Apply Newton’s laws, particularly Newton’s second law, to describe circular motion.

Background

Topic: Circular Motion and Newton's Laws

This question tests your ability to apply Newton's laws to objects moving in circles, focusing on centripetal force and acceleration.

Key Formula:

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

• Newton's Second Law: $F_{\text{net}} = m a_c$

Step-by-Step Guidance

1. Identify the net force required for circular motion (centripetal force).

2. Write the formula for centripetal acceleration.

3. Combine Newton's second law with the centripetal acceleration to express the net force.

Try setting up the equations before revealing the answer!

Q1. Define what is meant by a “resistive force”. What is the direction of a resistive force? What are the two models we will use to express the magnitude of the resistive force, and under what circumstances is each model relevant?

Background

Topic: Resistive Forces (Drag, Friction)

This question asks you to define resistive forces, their direction, and the mathematical models for their magnitude.

Key Models:

• Linear drag: $f_r = -b v$ (low speeds)

• Quadratic drag: $f_r = -c v^2$ (high speeds)

Step-by-Step Guidance

1. Define a resistive force and state its direction relative to motion.

2. Write the two models for resistive force and explain when each applies.

3. Discuss the physical meaning of the constants $b$ and $c$.

Try defining and distinguishing the models before revealing the answer!

Q2. Define “terminal speed” as it relates to the resistive force on a falling object. How can you calculate a relationship for terminal speed?

Background

Topic: Terminal Speed

This question tests your understanding of the balance of forces when an object falls through a resistive medium (like air).

Key Formula:

• At terminal speed: $mg = f_r$

• For linear drag: $v_t = \frac{mg}{b}$

• For quadratic drag: $v_t = \sqrt{\frac{mg}{c}}$

Step-by-Step Guidance

1. Define terminal speed and explain the physical situation when it occurs.

2. Set up the force balance equation for terminal speed.

3. Write the formula for terminal speed for both linear and quadratic drag cases.

Try setting up the equations before revealing the answer!

Q1. Understand what the scalar product of two vectors is: be able to describe it in words, and be able to connect this qualitative understanding to the mathematical formulas.

Background

Topic: Scalar (Dot) Product of Vectors

This question tests your understanding of the dot product, both conceptually and mathematically.

Key Formulas:

• Component form: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

• Magnitude-angle form: $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$

Step-by-Step Guidance

1. Describe in words what the scalar product represents (projection of one vector onto another).

2. Write both mathematical forms of the dot product.

3. Explain when each formula is most useful.

Try connecting the concept to the formulas before revealing the answer!

Q1. What is work? (Definition in words and mathematically, units, scalar or vector, positive or negative?)

Background

Topic: Work in Physics

This question asks you to define work, its mathematical expression, units, and properties.

Key Formula:

• $W = \vec{F} \cdot \vec{d}$

• Units: Joules (J)

Step-by-Step Guidance

1. Define work in words (force applied over a distance).

2. Write the mathematical formula for work (dot product of force and displacement).

3. State the units of work and whether it is a scalar or vector.

4. Discuss when work is positive or negative.

Try defining work and its properties before revealing the answer!

Q1. What is the spring/elastic force? Use the definition of work to find the work done by a spring. When is work done by a spring positive/negative? Extend this to the work done by other forces.

Background

Topic: Spring Force and Work

This question tests your understanding of Hooke's law, work done by a spring, and the sign of work for different forces.

Key Formulas:

• Hooke's Law: $F_s = -k x$

• Work by a spring: $W = \frac{1}{2} k x^2$ (with sign depending on direction)

Step-by-Step Guidance

1. State Hooke's law for the spring force and define each variable.

2. Set up the integral for work done by a variable force (the spring).

3. Evaluate the integral to find the work done by the spring.

4. Discuss when the work is positive or negative, and relate this to other forces.

Try setting up the integral and considering the sign before revealing the answer!

Q1. What is kinetic energy? What is potential energy? (in words and mathematical definitions)

Background

Topic: Kinetic and Potential Energy

This question asks you to define and write formulas for kinetic and potential energy, including gravitational and elastic forms.

Key Formulas:

• Kinetic energy: $K = \frac{1}{2} m v^2$

• Gravitational potential energy: $U_g = m g h$

• Elastic potential energy: $U_s = \frac{1}{2} k x^2$

Step-by-Step Guidance

1. Define kinetic energy in words and write its formula.

2. Define potential energy in words and write formulas for gravitational and elastic potential energy.

3. Explain what each variable represents.

Try writing the definitions and formulas before revealing the answer!

Q1. Define each of the following in words: Conservative force (examples?), Non-conservative force (example?), Mechanical energy.

Background

Topic: Types of Forces and Energy

This question tests your understanding of the difference between conservative and non-conservative forces, and the concept of mechanical energy.

Key Concepts:

• Conservative force: Path-independent work (e.g., gravity, spring force)

• Non-conservative force: Path-dependent work (e.g., friction)

• Mechanical energy: Sum of kinetic and potential energy

Step-by-Step Guidance

1. Define a conservative force and give examples.

2. Define a non-conservative force and give an example.

3. Define mechanical energy in terms of kinetic and potential energy.

Try defining each term before revealing the answer!

Q1. Describe what is meant by the statement “energy is conserved”. Create a mathematical statement of conservation of energy for specific systems and circumstances.

Background

Topic: Conservation of Energy

This question asks you to explain energy conservation and write equations for different types of systems.

Key Formulas:

• Isolated system: $E_{\text{initial}} = E_{\text{final}}$

• Only conservative forces: $K_i + U_i = K_f + U_f$

• With non-conservative forces: $K_i + U_i + W_{\text{nc}} = K_f + U_f$

Step-by-Step Guidance

1. Explain what it means for energy to be conserved in a system.

2. Write the general conservation of energy equation for an isolated system.

3. Adapt the equation for systems with only conservative forces, and for those with non-conservative forces.

Try writing the equations for each case before revealing the answer!

Q2. The “boundary” of the system can be defined by the problem-solver. Consider how the definition of the “boundary” of the system, and therein the definition of which forces are “external”, changes the mathematical statement of conservation of energy, but not the physics/answer.

Background

Topic: System Boundaries and Energy Conservation

This question asks you to consider how defining the system affects which forces are considered internal or external, and how this changes the energy equation.

Key Concepts:

• System boundary

• Internal vs. external forces

• Energy conservation equations

Step-by-Step Guidance

1. Explain what is meant by the boundary of a system in physics problems.

2. Discuss how changing the boundary changes which forces are considered external.

3. Describe how this affects the mathematical form of the conservation of energy equation.

Try considering different system boundaries before revealing the answer!

Q3. What is power? What are its units? What is the equation describing average power? What is the equation (and its derivation) for power delivered by a constant force?

Background

Topic: Power in Physics

This question asks you to define power, its units, and write equations for average and instantaneous power, including for a constant force.

Key Formulas:

• Average power: $P_{\text{avg}} = \frac{W}{\Delta t}$

• Instantaneous power: $P = \vec{F} \cdot \vec{v}$

Step-by-Step Guidance

1. Define power in words and state its SI units.

2. Write the formula for average power and explain each term.

3. Write the formula for power delivered by a constant force and show how it is derived from the definition of work and velocity.

Try writing the definitions and formulas before revealing the answer!