Below is the list of learning objectives for those exams you selected to see in the links above. Click on the "Show Details" link to get an idea and a few examples of what the learning objective is testing.
Math in relationship with Mechanics
- Be able to compute the components of a vector in any given coordinate system.
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The following are a few examples that fall inside this learning objective:
- Example 1: you are shown a 2D vector $\vec{A}$ drawn on a grid. Can you compute the components of the vector if provided with the angle with respect the y-axis and the magnitude of the vector ? What if the angle provided is with respect to the negative x-axis ?
- Example 2: you are shown a 2D vector $\vec{A}$ drawn on a grid. Can you compute the components of the vector if provided with the magnitude of the vector and the value of one of the components ?
- Example 3: you are shown a 2D vector $\vec{A}$ drawn on a grid that is marked such that you can measure distances on it. Can you give the components of that vector ?
- Be able to compute addition, scalar, and vector products between vectors.
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The following are a some examples that fall inside this learning objective:
- Example: you are given the components of two vectors $\vec{A}=(1,4,-5)$, and $\vec{B}=(-3,2,-4)$.
Can you compute the addition, scalar, and vector products between those vectors?
- Example: you are told that the magnitude of vector $\vec{A}$ is 13, the magnitude of $\vec{B}$ is 3 and the angle between those two vectors is 45 degree. Can you compute the magnitude of the cross and dot products between those two vectors ?
- Example: you are shown the vectors $\vec{A}$ and $\vec{B}$ in a plane. Can you identify the direction of the sum or subtraction of those vectors ? What about the general direction of the cross product $\vec{A} \otimes \vec{B}$ ?
- Be able to solve for a unknown quantity in a single equation when possible.
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If at some point in a problem you reach an equation in which you know the values of all variables except for one, you should be able to find out the value of the unknown variable from that equation. For example, if the statement of the problem tells you that you know the variables $F$, $\mu_k$, $m$, and $a$, and your problem leads you to the following equation
$$ F + N - \mu_{k} N = m a $$
Can you identify the unknown variable and find out its value in terms of the known variables ?
- Be able to solve a system of N equations with N unknown variables.
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If at some point in a problem you get some number of equations, say N, in which you know the values of all variables except for a number also N of them. Then you should be able to find out the value of all N unknown variables from that set of equations. For example, if the statement of the problem tells you that you know the variables $m_1$, $m_2$, $g$, and some angle $\alpha$, and you manage to have the two following equations
$$
\begin{align*}
\left\{ \begin{array}{l}
-T + m_2 g = m_2 a \\
+T + m_2 \sin(\alpha) = m_1 a
\end{array} \right.
\end{align*}
$$
Can you identify the two unknown variables and find out their values in term of the other known variables ?
In general, can you articulate what steps needs to be taken to solve a system of N equations with N unknowns ?
- Be able to identify and solve a quadratic equation.
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In a given problem you are told that the variables $L$ , $x_0$, $g$ and $v_0$ are known. Can you identify which of the following equations is a quadratic equation on an unknown variable and could you solve for such unknown variable ?
- $L = v_0 t + x_0^2 t $
- $ L = x_0 + v_0 t + \frac{1}{2} g t^2$
- $L^2 = v_0^2 t + x_0^2 t$
- Be able to translate verbal constraints into mathematical language.
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In this course there are many different constraints that will need translation into mathematical language and this learning objective targets the skills a student has in translating different constraints into mathematical language. As an example, imagine you have a problem in which you have obtained two functions that give the position of two objects as a function of time, for example $X_a(t)$ and $X_b(t)$. Assume now that the problem also states that the two objects collide at some unknown time. How would you translate that last statement into mathematical language ? Use common sense and think what is it meaning of that statement; in our case what does it mean to collide ? it means to be at the same place at the same time, and therefore in mathematical language we could write:
$$ X_a(t_c) = X_b(t_c), $$ where $t_c$ is the unknown time of the collision. Translating constraints into mathematical language typically gives you more equations with which we can solve the problem.
- Be able to translate mathematical results to verbal interpretations.
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The results of problems are often expressed in a numerical form and this learning objective tests the student's understanding of the implications of such result. For example after setting a coordinate system your results tells you that the component of the velocity in some direction is negative, what does that mean in terms of the movement of the object ? Another example would be to compute the necessary magnitude of a force for something to happen, and compare it with the maximum magnitude that the force can actually exert. What does it tell you when the former is larger than the latter ?
- Be able to do integrals, or take derivatives, of polynomials.
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Consider a function such as the one below $$ v(t) = v_0 + C t^3 $$, where $v_0$ and $C$ are known quantities of some value. Can you compute all of the things below ?
- the derivative of $v(t)$ with respect to time, or $\frac{d v(t)}{dt}$ ? Is this a function or a number ?
- the indefinite integral of $v(t)$ with respect to time, or $\int v(t)\, dt $ ? Is this a function or a number ?
- the definite integral of $v(t)$ with respect to time, from $t=4$ to $t=10$, or $\int_{4}^{10} v(t)\, dt $ ? Is this a function or a number ?
- the function obtained as the integral of $v(t)$ with respect to time, from $t=4$ to $t$, or $\int_{4}^{10} v(t)\, dt $ ? Is this a function or a number ?
Finally, do you know how derivatives of a function are obtained graphically when the function is plotted against the variable of derivation ?
Procedural and clarity of exposition
- Draw clearly a coordinate system when the solution to the problem involves a quantity that depends on it.
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Learn how to appropriately draw a right-handed Cartesian coordinate system. What is a right-handed coordinate system ?, can you identify when it is not ? It is very common for students to skip drawing a coordinate system or drawing it poorly, and then make mistakes when computing components of vectors on that coordinate system and getting signs wrong.
Units
- Be able to recall the meaning of most prefixes such as milli, kilo, mega, etc. Describe a methodology for unit-conversion and apply it to convert any quantity to the requested units.
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What is your methodology to do unit conversions ? Can you convert any quantity from one set of units to another ?. How about converting 13 $\frac{grams}{cm^2}$ to units of $\frac{kilograms}{meter^2}$ ?
Kinematics
- Average $\vec{r},\vec{v},\vec{a}$ : Explain the concept and definitions of average position, average velocity, and average acceleration and be able to compute them from basic quantities or measurements.
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From home I go to work every morning at 8am, drive 10 miles to work for about 12 minutes, and drive back home at 8pm, what is the average velocity between one day at 8am and the next day at 8am ?
- Instantaneous $\vec{r},\vec{v},\vec{a}$ : Explain the concept and definitions of instantaneous position, velocity, speed, and acceleration. Be able to compute them from an equation giving position vs time, and from a plot of position versus time. Be able to draw them in plots as a function of time. The same for angular quantities such as angular position, velocity, and acceleration.
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If you have an equation that tells position as a function time what can you do to obtain another function that gives you the velocity as a function of time ? What can you do to find the acceleration as a function of time ?
If instead you are given the position vs time in a graph, can you find the velocity at any given time?, what can you say about the acceleration at a given time?
- Direction of $\vec{v}$ and $\vec{a}$ : Be able to identify direction of velocity and general direction of acceleration at any point in time when given the trajectory of an object in space.
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Our daily life is so full of moving objects that we typically think nothing of them. This objective is about analyzing the movement of objects and being able to describe such movement in terms of the object's velocity and acceleration vectors. The following are examples that can help with that understanding by using your imagination. Close your eyes and imagine that you are at the top of a building and :
- you see a car moving down on the street while slowing down at a red light. In that mental picture attach a velocity and acceleration vectors that move with the car. Can you draw the direction of the velocity vector as the car reaches the light ?, is changing size ? what about the acceleration vector ? What if the light becomes green and the car accelerates again ?
- you see a car moving on the street and turning a corner to take another street all at constant speed, can you draw the direction of the velocity vector or the acceleration vector ?
- imagine a bird flying in circles at constant height, can you draw the direction of the velocity at any point ?
- imagine a small ball moving towards the right while falling and bouncing off the floor. Can you draw the direction of the vector velocity and acceleration at any position ?
- Equation of Motions : Be able explain the concept of ``equations of motion''. Be able to compute position and velocity as a function of time when given the acceleration as a function of time and initial velocity and position. The same for linear and angular position and velocity. Describe how that looks in mathematical form.
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What is an "equation of motion" ? It is simply one or several equations that describe the motion of an object. In this course they are typically position and velocity as a function of time. It is fundamental to understand all the terms in the equations; for example in the equations below what is $\vec{v}_0$, $\vec{x}_0$ and what is $t_0$ ? Are those related ?
$$
\vec{v}(t) = \vec{v}_0 + \int_{t_0}^t \vec{a}(t) dt \\
\vec{r}(t) = \vec{r}_0 + \int_{t_0}^t \vec{v}(t) dt
$$
- Constant vs non-constant acceleration : Be able to identify problems in which the acceleration is constant in time and those in which it is not.
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In many kinematic problems this is the first question that needs to be answered. Suppose a problem states that an object moves as described by the cases below, in which of those cases the object moves with constant acceleration ?
- with a position as a function of time given by $x(t)= c t + b t^2$
- with a position as a function of time given by $x(t)= c t + b t^3$
- with a velocity as a function of time given by $v(t)= c + b t$
- with a velocity as a function of time given by $v(t)= c + b t^2$
- with an acceleration as a function of time given by $a(t)= c + b t$
- with an acceleration of $a=c$ from the first 2 seconds and $a=2c$ after two seconds.
where $b$ and $c$ are non-zero known constants.
Circular Motion
- Circular Motion : Explain the concept of Circular Motion and be able to identify it as such.
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Circular motion is nothing else that motion on a circle. The moving object does not have to move at constant speed and it can even go back and forth on a circular path without ever completing the circle.
- Acceleration - Parallel compute the magnitude of the acceleration in the direction of the object's velocity based of the speed.
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Imagine an object moving on a circle and you freeze time at some time; can you draw the direction of the object's velocity at that time ? If not see LO 13, if you did then that is the direction of the velocity, but can you tell the component of the acceleration in the direction of the velocity at that point if provided with the speed as a function of time ?
Imagine the problem states that the speed as a function of time for the first 4 seconds is given by $s(t) = 1 - 0.4 t $ what is the component of the acceleration in the direction of the velocity at $t=2$ seconds ? Is it positive or negative ? what does it mean being positive or negative ?
Answer:
the component of the acceleration in the direction of velocity is given by $$ a_{\hat{v}} = \frac{d | \vec{v} |}{dt} $$ where the magnitude of the velocity is just the speed, so taking the derivative with respect to time we get $a_{v}= -0.4$. The negative sign indicates that the acceleration points OPPOSITE to the direction of the velocity.
- Acceleration - Perpendicular compute the direction and magnitude of the acceleration in the direction perpendicular to the velocity based of the speed and radius.
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As in the previous learning objective imagine an object moving on a circle and you freeze time at some time; can you draw the direction of the object's velocity at that time ? If not see LO 13, if you did then that is the direction of the velocity, but can you tell the direction and magnitude of the acceleration in the direction perpendicular to the velocity ?
Answer:
Yes, the magnitude is given by $$ a_{\perp} = \frac{v^2}{R} $$ where $v$ is the velocity at that time, and $R$ is the radius of the circle. The direction of this component is always pointing towards the center of the circle.
- Uniform Circular Motion : Identify what is UNIFORM Circular motion and be able to recognize it. For this motion recall the relationship between the speed of an object and the period and radius.
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Uniform Circular motion on a circle that is done under constant speed. If an object moves at constant speed $s$ and travels a distance equal to the circumference of a circle or radius $R$ in time $T$, then we have the relationship $$ s = \frac{2\pi R}{T} $$
Reference Frames
- Relatively Moving Reference Frames : Be able to translate position, velocity, and acceleration from one reference system to another reference system moving with constant velocity with respect to the first one. Show details
Consider the velocity of a plane with respect to ground, the velocity of that plane with respect to the air, and the velocity of the air with respect to ground. A typical problem gives two out of these three velocities and asks you to compute the other one.
Forces
- Newton's $1^{st}$ and $2^{nd}$ laws : Recall Newton's $1^{st}$ and $2^{nd}$ laws conceptually and mathematically. Be able to apply them to solve a problem.
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Answer:
Newton's first law postulates the existence of at least one reference frame relative to which the motion of a particle not subjected to forces is a straight line at a constant speed. These reference frames are called inertial or non-accelerated reference frames.
Newton's second law states that, when viewed from an inertial reference frame, the sum of the vector forces acting on an object of constant mass equals the mass times the acceleration. That is $$\sum_i \vec{F}_i = m \vec{a}.$$ This is a vector equation and thus valid component by component. That is $$\sum_i F_{x,i} = m a_x,$$ and equivalently for $y$ and $z$ components.
- Newton's $3^{nd}$ law : Recall Newton's $3^{rd}$ law conceptually. Be able to identify action-reaction pair of forces in a problem.
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Answer:
Newton's third law states that in the interaction between two bodies, when viewed from an inertial reference frame, when one body exerts a force on a second body the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. These two forces are associated to the same interaction and said to be a pair of action-reaction forces.
Question:
The moon exerts a gravitation pull force on an astronaut in its surface, what is the action-reaction of the force that the astronaut feels ?
- Weight : define the concept of Weight and write it mathematically and be able to explain the origin of the weight force. Indicate in what direction it acts and be able to draw it in a free-body diagram.
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Answer:
Weight is the force of gravity between two objects. Typically the weight of an object or person on the surface of the planet is given by $W = m g$ where $m$ is the mass of the object and $g$ is the gravitational acceleration on the surface of that planet. The direction of the weight force is towards the planet.
- Tension : recognize the conditions in which the magnitude of the tensions in either side of a String/Rope/Chain are the same, or the conditions in which they are not. Recognize the conditions in which the tensions in either side of a String/Rope/Chain around a pulley are the same, or the conditions in which they are not. Indicate the direction of the tension force makes on object and be able to draw the tension in a free-body diagram.
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Answer:
The tension in a String/Rope/Chain is the same in one end as in the other if the String/Rope/Chain is massless.
The tensions that a String/Rope/Chain applies on either side of a pulley are the same in both ends when the pulley has zero moment of inertia. If the pulley has no mass then its moment of inertia is also zero, and the tensions are the same. PUT PICTURE HERE.
- Spring force : explain how the force of a spring behaves as it is compressed/expanded and recall the mathematical formulation of such force in terms of the relevant parameters.
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Answer:
A spring has a natural length $l_0$ at which is nor expanded nor compressed. The string is compressed or expanded if the size of the string is smaller or larger than the natural length respectively. If a string has one end pinned at $x=0$ and the other attached to an object at position $x$, the force the spring is exerting on the object is given by $F_s(x) = -k (x-{l_0})$ where $k$ is the constant of the spring. The minus sign in the equation refers to the fact that the force points in the negative $\hat{x}$ direction when x is larger than $l_0$ which makes sense since the spring is expanded and wants to bring the object back to it's natural equilibrium position at $l_0$.
- Normal : identify what are normal forces and indicate in what direction they act. Be able to draw them in a free-body diagram.
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Answer:
Normal forces are forces that have a direction that is perpendicular to the local surface of contact between the two objects.
- Static vs Kinetic Friction : Be able to identify whether there is kinetic or static friction force between two objects in contact.
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Answer:
If the two objects in contact are moving
relative to each other there is kinetic friction force. If they are rest
relative to each other then it might be static friction.
- Kinetic Friction : recall the relation between the kinetic friction force, the coefficient of kinetic friction, and the normal force. Identify the direction of the kinetic friction force in both of the objects making contact. Identify what components determine the value of the coefficient of kinetic friction. Be able to draw it in a free-body diagram.
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Answer:
The magnitude of the kinetic friction force is always given by $f_k=\mu_k N$ where $\mu_k$ is the coefficient of kinetic friction and $N$ is the magnitude of the normal force between the two objects. The direction of this friction force is such that it goes against their relative velocity. The value of the coefficient of kinetic friction depends only on the type of the surface of contact and the materials with which both objects are made of.
- Static Friction : recall the relation between the static friction force, the coefficient of kinetic friction, and the normal force. Identify the direction of the static friction force in both of the objects making contact. Identify what components determine the value of the coefficient of static friction. Be able to draw it in a free-body diagram.
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Answer:
The direction of the static friction force goes against the direction of movement the object will have if the static friction force wasn't there. The magnitude of the static friction force is exactly what it needs to be so that there is no movement. However the
maximum static friction force that could exist between two objects is determined by $f_{s} =\mu_s N$ where $\mu_s$ is the coefficient of static friction and $N$ is the magnitude of the normal force between the two objects. The value of the coefficient of static friction depends only on the type of the surface of contact and the materials with which both objects are made of.
- Other Forces: Be able to identify external forces (other than Weight, Tension, force of spring, Normal or frictional forces) and be able to draw them in a free-body diagram.
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This objective requires the identification of other, less common, forces acting on objects. For example, a plane drops a package while flying and you are told the wind exerts a horizontal force on the package as it falls. You are asked to draw a free body diagram of the package and you would need to include the force of the wind to pass this objective. Missing a force from a free-body diagram will result in the wrong answer.
- Equilibrium vs Dynamic: Categorize and distinguish problems that are of dynamic nature vs those that are of in equilibrium. Explain what steps should be followed to solve either of these type of problems.
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Imagine you have a system that contains many objects that may be moving. When is that system considered to be in equilibrium and when is not ?
- Work: Define the work done by a force while applied on a moving object. Be able to compute it if provided with the force that changes as a function of position.
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If somebody asks you to directly measure "distance" the first question you have is between which points. Why ?, because you understand the definition of distance requires you to know the relative location of two points. If somebody asks you to directly compute "work" what questions should you have ? What does the definition of work entails you know ?
- Power: Define instantaneous and average power in terms of work done and the time it takes to do it. Write instantaneous power in terms of the force and the velocity of the object. Be able to compute power when needed.
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Example 1: Two liftforks from different brands lift a crate from the floor placing them in a shelf 2 meters above ground. One device takes longer time than the other. Which has the bigger power ? Can you estimate what the average power was ?
Example 2: A block is moving with velocity $v_0$ exactly in the $\hat{x}$ direction when a force of magnitude $\vec{F}=(5,7)$ is applied on it. What is the power done by the Force
Energy
- Kinetic - Translational : Recognize when an object has translational kinetic energy, recall its mathematical formulation, and be able to apply it to the solution of problems.
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An small car is moving with respect to you, what is the translational kinetic energy of the car with respect to you ? What is the translational kinetic energy of the car with respect to the driver of the car ?
- Kinetic - Rotational : Recognize when an object has rotational kinetic energy, recall its mathematical formulation, and be able to apply it to the solution of problems.
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A helicopter is hovering in front of you, what is the rotational energy of the blades ?
- Conservative vs Non-conservative : Explain the definition of a conservative force. Distinguish between conservative and non-conservative forces.
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Imagine a new force is found in nature, how should that force behave so that you know it is conservative or non-conservative ?
- Potential : Identify those forces that can have an associated potential energy. Recall the general mathematical relationship between a conservative force and its associated potential energy. Be able to compute one from the other when possible.
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What does the word "potential" means ? Can all the forces have an associated potential energy ?
If you are told that an object moves in three dimensions under a force whose potential energy can be written as $U(x,y,z)=4 x^2 + 4 y^2 + 3 z^2$ in some coordinate system. Can you compute the three components of the force applied on the object for that coordinate system ?
- Gravitational and Spring Potential: Recall the mathematical formulation for the gravitational potential on the surface of a planet and for the ideal spring elastic force.
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What does the word "potential" means ? Can all the forces have an associated potential energy ?
- Work-Energy Theorem : Explain the concept of the work-energy theorem, recall its mathematical formulation, and identify all pieces of it. Be able to recognize problems in which it may be applied. Explain the derived concept of the conservation of energy theorem. Be able to recall its mathematical formulation and identify all pieces of it. Be able to recognize problems in which it may be applied.
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It is called the "work-energy" theorem, so it relates some "work" to some "energy", but most precisely "work of all forces applied on an object as it goes from one point to another" to "change of kinetic energy of the object as it goes moves from that same initial point to the same final point". Think about this equation, does it directly uses the acceleration of the object ? or the velocity ?, or the position ?.
- Mechanical Energy : Recall the definition of mechanical energy and be able to explain in what conditions is the mechanical energy conserved. Be able to recognize problems in which conservation of mechanical energy might be of use for their solution.
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What is mechanical energy ? What equation tells you the conditions in which the mechanical energy is conserved ?
Energy Diagrams
- Energy Diagram : Define what is an energy diagram. Be able to compute the kinetic and potential energy at any point in an energy diagram when possible.
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An energy diagram shows the energy of an object in the y-axis vs versus position. The graph might contain curves for potential energy, mechanical energy, kinetic energy, or combinations of those.
If you are given an energy diagram that shows both the mechanical energy and the potential energy of an object as a function of its position. When the object is at a given position, can you tell its potential, mechanical and even kinetic energy ?
- Equilibrium Points : Identify equilibrium points in the energy diagrams and categorize whether they are stable and unstable.
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Can you get equilibrium points from a plot showing just potential energy versus position ? what about from a plot showing just mechanical energy ?
- Range of Movement : Be able to identify the maximum range of movement from an energy diagram.
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You are given a diagram showing an object's potential energy versus its position as well as the mechanical energy of the object. You are also told what the initial position of the object is. Can you tell what is the maximum positions the object may reach ?
- Forces from diagrams : Be able to compute forces from an potential-energy vs position diagram.
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What is the relationship between potential-energy and forces ? How is that translated mathematically ? See LO 44.
Center of Mass
- Center of Mass : Identify the position/velocity/acceleration of the center of mass for a system of objects given their masses and positions/velocities/accelerations.
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What is the definition of the center of mass position ? How do you get velocity from position ?.
Linear Momentum
- Linear Momentum : Explain the concept of linear momentum. Describe and contrast the linear momentum definitions of a point-like particle, of a system of particles, and of a solid. Be able to compute them from their definitions when possible.
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Let's see a few typical cases:
- You have two objects of masses $m_1$ and $m_2$ moving with velocities $\vec{v}_1$ and $\vec{v}_2$ respectively, what is the linear momentum of object 1 ?, what is the linear momentum of the system composed of object 1 and 2 ?
- Now you have a big object, say a cylinder, that rotates at some angular velocity $\omega_{cyl}$ around some axis and at the same time its center of mass moves at some velocity $\vec{v}_{cyl}$. What is its linear momentum ?
- Relationship with Forces : Explain the relationship of the linear momentum of a system with the forces acting on it.
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You have a system of particles with different forces, some of them internal and some external to the system. What is the relationship of these forces to the linear momentum of the system ? Can anything be said about an individual particle in the system ?
- Conservation : Describe in what conditions it the linear momentum of a system conserved, and be able to recognize problems in which its conservation might be of use for their solution.
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You are solving a problem in an exam in which you have a system of particles and you know all the forces that act upon all particles in your system. When is the linear momentum of the system conserved ?
- Impulse: Define the concept of impulse of a force and the nature of all quantities in the mathematical formulation. Identify its relationship with changes on the linear momentum of the object to which the force is applied. Be able to compute and use impulse to solve problems.
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A force that changes with time $\vec{F}(t)= (\alpha t^2 - \beta ) \hat{x}$ (where $\alpha$ and $\beta$ are just constants) is applied on an object from $t=2$ to $t=5$. Can you compute the impulse given by the force to the object during that time ? If you were given the initial momentum of the object at $t=2$ could you find the momentum at $t=5$ ?
Collisions
- Classification : Be able to classify collisions into elastic, inelastic, and completely inelastic categories. Be able to include this information to solve a collision problem when possible.
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In a problem you are told that an elastic collision occurs between two objects. What equation you know you can use because of that ?
Moment of Inertia
- Moment of Inertia : Articulate the concept of moment of inertia and recall its definition with respect to an arbitrary axis for a point-like particle and for a system of point-like particles. Be able to articulate the nature of all quantities in the mathematical formulation.
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In a few words can you explain what is the moment of inertia ?
- A single point-like particle of mass $m$ is attached perpendicularly to a rod of length $L$ whose other end is attached to an axis. What is the moment of inertia of the point-like particle with respect to the axis ? What if the rod is not perpendicular to the axis ?
- You have two point-like objects of masses $m_1$ and $m_2$, each attached to a perpendicular road going to the same axis. If the roads have length $L_1$ and "L_2$ respectively, what is the moment of inertia of the system of two masses around that axis ?
- Parallel axis theorem : Explain the parallel axis theorem, its usefulness, and be able to apply it in problems to compute the moment of inertia around any axes when necessary. Be able to articulate the nature of all quantities in the mathematical formulation.
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The parallel-axis theorem relates what to what ? Why does the theorem has the words "parallel-axis" in the title, what axes need to be parallel ?
- Composite Objects : Be able to compute the moment of inertia of single object based of the moment of inertia of its constituents parts.
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Imagine you have small disk glued off-center on another bigger disk. Can you compute the moment of inertia of the glued system with respect to the center of the big disk if provided with a the moment of inertia of both disks with respect to their own center of masses ?
Torques
- Torque : Recall the definition of torque and the nature of all quantities in the mathematical formulation. Be able to compute it from basic quantities.
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In a few words can you explain what is the torque of a force ? Is it a vector ? and if so what is its direction ? Does the torque depends on a point or an axis ? Imagine you have a big disk of radius $R$ over which there is a force $\vec{F}$ acting somewhere on the surface of the disk. Can you compute the torque of that force with respect to the center of the disk ? do you have all the information you need to compute the torque ? If not what else do you need ?
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The torque is the effectiveness of a force to generate a rotation around a point and is a vector defined as $\tau_q = \vec{r} \times \vec{F}$ where $q$ is the point on space around which we compute the torque and $\vec{r}$ is the distance between $q$ and the position on the disk where the force acts on. Since we don't know where on the disk the force is acting we can't compute the torque of such force.
- Relationship with angular acceleration : Explain the relationship of net torque with the moment of inertia and angular acceleration of an object. Be able to apply this relationship to solve problems.
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Imagine the same problem stated in the previous objective: an object in space is rotating around some axis. Some forces are applied on the object such that the angular momentum keeps its direction but its magnitude increases at a constant rate of $3 \frac{kg m^2}{s}$ every minute. What do you need to know to compute the angular acceleration of the object ? Around what axis ?
- Work of a Torque : Be able to compute the work done by a torque.
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What is the definition of work of a torque ? How is this related to the work of a force ? Imagine a wheel is spinning with constant velocity around its center axis and a motor produces a torque of magnitude $\tau$ on this axis speeding the wheel up. What is the work of the torque as the wheel makes one full revolution ?
Angular Momentum
- Angular Momentum : Compute the angular momentum of a an object moving translationally, one at rest but spinning around its center of mass, and one doing both. Explain if any case depends on the point of reference.
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Imagine you are an astronaut in space and you see an asteroid the size of a car pass nearby you. What quantities do you need to compute the angular momentum of that asteroid with respect to you when:
- the asteroid moves with velocity of $\vec{v}_{CM}$ but does not rotate.
- the asteroid does not move translationally with respect to you, but rotates with angular velocity $\vec{\omega}$ around its center of mass.
- the asteroid moves with velocity of $\vec{v}_{CM}$ and rotates around its center of mass with angular velocity $\vec{\omega}$.
- Relationship with Torques : Explain the relationship between the net torque on a system and its angular momentum and be able to articulate the nature of all quantities in the mathematical formulation. Be able to apply this relationship to solve problems.
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An object in space is rotating around some axis. Some forces are applied on the object such that the angular momentum keeps its direction but its magnitude increases at a constant rate of $3 \frac{kg m^2}{s}$ every minute. What can you tell about the torques on the object ? what about the direction ?
- Conservation : Describe in what conditions is the angular momentum conserved and be able to recognize problems in which its conservation might be used to find solutions.
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When is the angular momentum of a system conserved in a given direction ? When the torque of the external forces in that direction is zero.
Gravitation
- Gravitation : Describe the nature of the gravitational force and the meaning of all quantities in its mathematical formulation. Explain and identify the conditions in which the mathematical formulation is valid.
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The magnitude of the gravitational force between two point-like particle is expressed by a formula. What are the different components on that formula ? Can that formula be used to accurately compute the gravitational attraction between two cubes of size $L$ and mass $m$ separated by a distance of $2L$ in space ?
- Potential : Recall the typical formula for the gravitational potential energy and reproduce the shape it would have in an energy-position diagram.
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Typically an energy diagram of the gravitation potential has position on the horizontal axis. In the example of the earth orbiting around the sun, what is that position at any point ?
- Gravitational bounds : Explain the concept of having an object ``gravitationally bound'' to another. Be able to determine if an object is gravitationally bound to another. Similarly explain the concept of ``escape velocity'' and be able to compute it in problems.
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A spacecraft trying to escape the gravitational attraction of a planet runs out of fuel when it is a distance $d$ from the center of the planet and has a velocity of $v_d$. What calculation you need to do to know if this spacecraft will fall back to the planet or escape its gravity ?
- Kepler's Laws : Briefly describe Kepler's Laws. Be able to articulate the nature of all quantities in the mathematical formulation.
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There are three laws of Kepler, can you briefly explain them ?, can you relate some of those as coming from conservation of other quantities ?
- Ellipse : Explain the main characteristics of an ellipse and be able to compute relevant distances from other distances in a problem.
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Draw an ellipse, identify the focii points and the minor and major axes. Identify the distance between the geometrical center of the ellipse and the focii points in terms of the eccentricity of the ellipse.
Periodic Motion
- Variables of Periodic Motion : Identify typical variables used to describe periodic motion, and explain the relationships between them.
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- Simple Harmonic Motion : Indicate the type of forces that results in simple harmonic motion. Be able to identify the equation of simple harmonic motion, compute the angular frequency from that equation, and recall the mathematical solution to it. Be able to apply this knowledge to the solution of problems.
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In a given problem somebody tells you an object is subjected to several forces. How should those forces behave so that you know that the object will move in simple harmonic motion ? How would you go about finding the frequency of the oscillation ? What do you need to know to be able to find the amplitude and phase of the movement ?