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:

• First Law (Law of Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force.

• Second Law: The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass.

• Third Law: For every action, there is an equal and opposite reaction.

Step-by-Step Guidance

1. Write out each law in your own words, focusing on the main idea (inertia, force and acceleration, action-reaction).

2. Think of a simple example for each law (e.g., a book at rest, pushing a cart, a rocket launch).

3. Consider how these laws apply to everyday situations and how they relate to each other.

Try summarizing 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 Formulation)

This question asks you to express Newton's second law in mathematical form, both as a vector equation and in one dimension, and to relate the magnitudes.

Key Formula:

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

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

• Magnitude relationship: $F_{\text{net}} = m a$

Step-by-Step Guidance

1. Recall that force and acceleration are both vectors, so the most general form uses vector notation.

2. For motion along a single axis (e.g., x-axis), write the equation for that component only.

3. When considering only magnitudes (and when force and acceleration are in the same direction), drop the vector notation.

Try writing each form of the equation 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 real-life situations, demonstrating your understanding of their practical implications.

Key Concepts:

• First Law: Inertia, objects at rest/motion, net force.

• Third Law: Action-reaction pairs, interactions between objects.

Step-by-Step Guidance

1. Think of a situation where an object remains at rest or moves at constant velocity unless acted on (e.g., a puck sliding on ice).

2. For the third law, consider interactions where two objects exert forces on each other (e.g., jumping off a boat, walking).

3. Describe what the action and reaction forces are in your example.

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 difference between mass (a measure of matter) and weight (the force due to gravity).

Key Terms and Formulas:

• Mass ($m$): A measure of the amount of matter in an object (scalar, SI unit: kg).

• Weight ($W$): The gravitational force acting on an object (vector, SI unit: N).

• Relationship: $W = m g$

Step-by-Step Guidance

1. Define mass and weight in your own words, noting their units and whether they are scalar or vector quantities.

2. Write the formula relating weight and mass, identifying what $g$ represents (acceleration due to gravity).

3. Explain how mass is intrinsic to the object, while weight depends on the gravitational field.

Try explaining the difference in your own words 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: Named 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: Downward force due to Earth's mass.

• Normal: Perpendicular contact force from a surface.

• Tension: Pulling force transmitted by a string/rope.

• Friction: Force opposing relative motion between surfaces.

Step-by-Step Guidance

1. For each force, describe its origin and direction (e.g., gravity always down, normal always perpendicular to surface).

2. State when each force should be included in a free-body diagram (e.g., normal force when in contact with a surface).

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

Try describing each force and its properties 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 apply Newton's second law.

Key Concepts:

• Free body diagram: Visual representation of all forces acting on an object.

• Frame of reference: Choice of axes (x, y, etc.) for analysis.

• Newton's second law: $\sum F = m a$ (in chosen direction).

Step-by-Step Guidance

1. Draw the object and represent all forces acting on it with arrows.

2. Choose a convenient coordinate system (e.g., align x-axis with motion or incline).

3. Write Newton's second law for each direction, summing forces along each axis.

Try drawing a free body diagram for a simple situation before checking the answer!

Q1 (Chapter 13). 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. Identify the two masses involved: the object ($m$) and Earth ($M_E$).

2. Write the universal gravitation formula for these two masses, with $r$ as Earth's radius.

3. Relate this to the familiar $F = m g$ by recognizing $g = G \frac{M_E}{R_E^2}$.

Try connecting the universal gravitation formula to the weight formula before checking the answer!

Q2 (Chapter 13). Describe “Universal Gravitation”:

Background

Topic: Newton's Law of Universal Gravitation

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

Key Formula:

• Direction: Always attractive, along the line joining the centers of mass.

• Magnitude: $F = G \frac{m_1 m_2}{r^2}$

Step-by-Step Guidance

1. State that gravity acts along the line connecting the centers of the two masses.

2. Write the formula for the magnitude, identifying each variable ($G$, $m_1$, $m_2$, $r$).

3. Discuss that the force is always attractive and acts equally on both masses.

Try explaining the direction and magnitude in your own words before checking the answer!

Q1 (Chapter 6.1). 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 a circle, focusing on the concept of centripetal force.

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. Recognize that an object in circular motion experiences a net force toward the center of the circle.

2. Write the expression for centripetal acceleration.

3. Apply Newton's second law to relate the net force to the mass and centripetal acceleration.

Try writing out the force equation for circular motion before checking the answer!

Q1 (Chapter 6.2). 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)

This question asks you to define resistive forces, their direction, and the two common models for their magnitude (linear and quadratic drag).

Key Formulas:

• Linear drag: $F_r = -b v$ (low speeds, small objects in viscous fluids)

• Quadratic drag: $F_r = -c v^2$ (high speeds, large objects in air)

Step-by-Step Guidance

1. Define a resistive force as one that opposes the motion of an object through a medium.

2. State that the direction is always opposite to the velocity.

3. Describe the two models and when each applies (linear for slow, quadratic for fast motion).

Try matching each model to a real-world example before checking the answer!

Q2 (Chapter 6.2). 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 concept of terminal speed and how to derive its formula using force balance.

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 as the constant speed where the upward resistive force equals the downward gravitational force.

2. Set up the force balance equation: $mg = F_r$.

3. Solve for $v_t$ using the appropriate model (linear or quadratic drag).

Try setting up the force balance for a falling object before checking the answer!

Q1 (Chapter 7.1). 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

This question asks you to explain the dot product both conceptually and mathematically.

Key Formulas:

• Definition: $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$

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

Step-by-Step Guidance

1. Describe the dot product as a way to multiply two vectors to get a scalar.

2. Write the geometric definition involving the cosine of the angle between vectors.

3. Write the component formula and explain when to use each.

Try connecting the two formulas before checking the answer!

Q1 (Chapter 7.2). What is work?

Background

Topic: Work

This question asks you to define work in words and mathematically, state its units, and discuss its properties.

Key Formula:

• Definition: $W = \vec{F} \cdot \vec{d}$

• Units: Joules (J)

• Work is a scalar quantity.

Step-by-Step Guidance

1. Define work as the transfer of energy via force acting over a distance.

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

3. State the units and whether work can be positive or negative.

Try writing the definition and formula before checking the answer!

Q1 (Spring Force). What is the spring/elastic force?

Background

Topic: Spring (Elastic) Force

This question asks you to define the force exerted by a spring and its mathematical expression (Hooke's Law).

Key Formula:

• Hooke's Law: $F_s = -k x$

Step-by-Step Guidance

1. State that the spring force is a restoring force, proportional to displacement from equilibrium.

2. Write Hooke's Law, identifying $k$ (spring constant) and $x$ (displacement).

3. Explain the negative sign (force opposes displacement).

Try writing Hooke's Law and explaining each term before checking the answer!

Q1 (Energy). What is kinetic energy? What is potential energy?

Background

Topic: Kinetic and Potential Energy

This question asks you to define kinetic and potential energy, both in words and mathematically.

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 as energy of motion, and write its formula.

2. Define potential energy as stored energy due to position or configuration.

3. Write formulas for gravitational and elastic potential energy.

Try writing the definitions and formulas before checking the answer!

Q1 (Chapter 8). 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 (isolated, conservative, non-conservative).

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. State that energy cannot be created or destroyed, only transformed.

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

3. Modify the equation for systems with only conservative or with non-conservative forces.

Try writing the energy conservation equation for a simple system before checking the answer!

Q3 (Chapter 8). 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

This question asks you to define power, state its units, and write equations for average and instantaneous power.

Key Formulas:

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

• Instantaneous power (constant force): $P = \vec{F} \cdot \vec{v}$

• Units: Watts (W) = Joules/second (J/s)

Step-by-Step Guidance

1. Define power as the rate at which work is done or energy is transferred.

2. Write the formula for average power in terms of work and time.

3. Write the formula for instantaneous power delivered by a constant force.

Try writing the power equations before checking the answer!