Guided Study: 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 that govern classical mechanics. Newton's three laws describe how objects move and interact with forces.

Key Terms:

• 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 key 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, or two skaters pushing off each other).

3. Consider how these laws apply to both stationary and moving objects.

Try summarizing each law in your own words before checking the textbook or notes!

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 different mathematical forms, considering both vectors and magnitudes.

Key Formulas:

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

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

• Magnitude relationship (when force and acceleration are in the same direction): $F_{\text{net}} = m a$

Step-by-Step Guidance

1. Recall that Newton's second law relates net force, mass, and acceleration.

2. Write the law in vector form, showing that both force and acceleration are vectors.

3. For motion along a single axis (e.g., x-axis), simplify the vector equation to one dimension.

4. When force and acceleration are parallel, express the relationship using only magnitudes.

Try writing each form of the equation before moving on!

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, helping you understand their practical significance.

Key Concepts:

• First Law (Inertia): Objects resist changes in motion.

• Third Law (Action-Reaction): Forces always come in pairs.

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, the boat moves backward).

3. Describe each scenario, identifying the forces involved and how they illustrate the law.

Try to come up with your own examples before looking up standard ones!

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 Formula:

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

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

• Relationship: $W = m g$

Step-by-Step Guidance

1. Define mass and weight in your own words, noting their units.

2. Write the formula relating weight and mass, identifying $g$ as the acceleration due to gravity.

3. Explain how mass is an intrinsic property, while weight depends on location (e.g., Earth vs. Moon).

Try explaining the difference in your own words before checking the definitions!

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

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 or rope.

• Friction: Opposes relative motion between surfaces.

Step-by-Step Guidance

1. For each force, describe what it is and when it acts (e.g., normal force acts when in contact with a surface).

2. State whether the magnitude is constant or variable (e.g., gravity is usually constant near Earth's surface, friction depends on surfaces).

3. Describe the direction of each force (e.g., normal is always perpendicular to the surface).

Try listing the properties of each force before reviewing your notes!

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 focuses on the use of diagrams to visualize forces and the importance of choosing a coordinate system.

Key Concepts:

• Free Body Diagram (FBD): A sketch showing all forces acting on an object.

• Frame of Reference: The coordinate system used to analyze motion.

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

Step-by-Step Guidance

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

2. Choose a coordinate system (e.g., x and y axes) that simplifies the problem.

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

Try drawing a free body diagram for a simple scenario before moving on!

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 gravity on Earth has its observed value, connecting it to Newton's law of universal gravitation.

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. Recall that gravity is the force between two masses (Earth and object).

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

3. Explain how, at Earth's surface, this leads to the familiar $g \approx 9.8$ m/s$^2$.

Try connecting the universal gravitation formula to the value of $g$ on Earth!

Q2. Describe “Universal Gravitation”: a) What is the direction of the force of gravity between any two objects having mass? b) 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 describe both the direction and magnitude of gravitational force between 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 two masses and is always attractive.

2. Write the formula for the magnitude, identifying $G$, $m_1$, $m_2$, and $r$.

3. Explain how the force depends on the product of the masses and the inverse square of the distance.

Try describing the direction and writing the formula before checking your 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 the concept of centripetal force.

Key Formula:

• Centripetal force: $F_c = m \frac{v^2}{r}$

• Newton's second law: $\sum F = m a$

Step-by-Step Guidance

1. Recognize that an object in circular motion experiences acceleration toward the center (centripetal acceleration).

2. Apply Newton's second law to the radial direction: $\sum F_{\text{radial}} = m a_c$.

3. Express centripetal acceleration as $a_c = \frac{v^2}{r}$ and substitute into the force equation.

Try applying these steps to a specific example, like a car turning in a circle!

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, Air Resistance)

This question asks you to define resistive forces and distinguish between linear and quadratic models.

Key Concepts and Formulas:

• Resistive force: Opposes motion, acts opposite to velocity.

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

• Quadratic model: $F_r = -c v^2$ (high speeds, large objects, turbulent flow)

Step-by-Step Guidance

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

2. Describe the linear model and when it applies.

3. Describe the quadratic model and when it applies.

Try to match each model to a real-world example!

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 focuses on the concept of terminal speed, where the resistive force balances gravity.

Key Formula:

• At terminal speed: $F_{\text{gravity}} + F_{\text{resistive}} = 0$

• For quadratic drag: $mg = c v_t^2$

• Solve for terminal speed: $v_t = \sqrt{\frac{mg}{c}}$

Step-by-Step Guidance

1. Define terminal speed as the constant speed when net force is zero.

2. Set up the force balance equation: gravity equals resistive force.

3. For a given model (e.g., quadratic), write the equation and solve for $v_t$.

Try deriving the terminal speed formula for both linear and quadratic drag!

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

This question tests your understanding of 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 formula involving magnitudes and the angle between vectors.

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

Try connecting the geometric and component forms of the dot product!

Q2. Express two mathematical formulas for the dot product (scalar product): as with all equations, make sure you can define what each symbol/variable in each formula represents, when each is applicable, and be able to move back and forth between each formula.

Background

Topic: Dot Product Formulas

This question asks you to write both the geometric and component forms of the dot product and understand their equivalence.

Key Formulas:

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

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

Step-by-Step Guidance

1. Write both formulas and define each variable ($A_x$, $B_x$, $\theta$, etc.).

2. Explain when to use each formula (geometric for angle, component for coordinates).

3. Describe how the two forms are equivalent.

Try writing out both forms and matching variables to physical meaning!

Q1. What is work? a) Definition (in words and mathematically). b) Units. c) Scalar or vector? d) Positive or negative?

Background

Topic: Work

This question tests your understanding of the concept of work in physics, including its definition, units, and properties.

Key Formula:

• Work: $W = \vec{F} \cdot \vec{d} = F d \cos \theta$

• Units: Joules (J)

Step-by-Step Guidance

1. Define work in words (force causing displacement).

2. Write the mathematical formula and identify each variable.

3. State the SI unit and whether work is a scalar or vector.

4. Discuss when work is positive or negative (based on angle between force and displacement).

Try answering each part before checking your notes!

Q1. What is the spring/elastic force? 2) Use the definition of work to find the work done by a spring. a) When is work done by a spring positive/negative? Extend this to the work done by other forces. 3) Use the definition of work and the work done by a spring to find the work done by an external force acting against the spring. Extend this understanding to the work done by external forces acting against other forces (e.g. gravity or friction).

Background

Topic: Spring Force and Work

This question explores Hooke's law, work done by springs, and the concept of external forces doing work against other forces.

Key Formulas:

• Hooke's Law: $F_s = -k x$

• Work by spring: $W = \frac{1}{2} k x^2$ (for stretching/compressing from equilibrium)

Step-by-Step Guidance

1. Define the spring force and its direction (restoring force).

2. Use the work formula to find work done by the spring as it is stretched or compressed.

3. Discuss when the work is positive or negative (depends on direction of force and displacement).

4. Apply the same reasoning to external forces acting against the spring or other forces.

Try deriving the work done by a spring using integration if you know calculus!

Q1. What is kinetic energy? (in words and a mathematical definition) 2) What is potential energy? (in words and a mathematical definition) a) Gravitational potential energy. b) Elastic potential energy.

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 and write its formula.

2. Define potential energy and distinguish between gravitational and elastic forms.

3. Write the formulas for each type and identify variables.

Try to explain the physical meaning of each type of energy!

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

Background

Topic: Types of Forces and Energy

This question asks you to define conservative and non-conservative forces and mechanical energy, with examples.

Key Concepts:

• Conservative force: Work done is path-independent (e.g., gravity, spring force).

• Non-conservative force: Work depends on path (e.g., friction).

• Mechanical energy: Sum of kinetic and potential energy.

Step-by-Step Guidance

1. Define each term in your own words.

2. Give at least one example for each type of force.

3. Explain what is included in mechanical energy.

Try to think of real-world examples for each type of force!

Q1. Describe what is meant by the statement “energy is conserved”. Create a mathematical statement of conservation of energy for specific systems and circumstances. a) “isolated” systems b) Systems in/on which only conservative forces act c) Systems in/on which non-conservative forces act

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. State what it means for energy to be conserved (total energy remains constant).

2. Write the conservation equation for an isolated system.

3. Modify the equation for systems with only conservative forces, and then for systems with non-conservative forces (include work done by non-conservative forces).

Try writing the equations for each case before checking your notes!

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 explores how defining the system boundary affects which forces are considered internal or external, and how this changes the energy equation.

Key Concepts:

• System boundary: Determines which forces are internal/external.

• External forces do work that changes the system's energy.

Step-by-Step Guidance

1. Define the system and identify which forces are internal and which are external.

2. Write the energy conservation equation, including work by external forces if necessary.

3. Explain that the physics (total energy change) is the same, but the equation's form depends on your system definition.

Try drawing a diagram to visualize the system boundary!

Q3. What is power? a) Units? b) Equation describing average power. c) 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 of doing work.

2. Write the formula for average power and identify each variable.

3. For a constant force, derive the formula for instantaneous power using the dot product of force and velocity.

Try deriving the instantaneous power formula from the definition of work!