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Physics I - Mechanics

• Course name: Physics I - Mechanics
• Course number: XYZ

Course Characteristics

Key concepts of the class

• fundamental concepts of physics for calculating problems of mechanics in \begin{itemize}
• statics,
• dynamics.

What is the purpose of this course?

This course provides the fundamental concepts of physics, in particular focusing on classical mechanics. In general, the aim of this course is: \begin{itemize} \item to study physical phenomena and laws of physics, the limits of their applicability, application of laws in the most important practical applications; to get acquainted with the basic physical quantities, to know their definition, meaning, methods and units of their measurement; to imagine the fundamental physical experiments and their role in the development of science; to know the purpose and principles of the most important physical devices; \item to acquire skills of work with devices and equipment of modern physical laboratory; skills of use of various methods of physical measurements and processing of experimental data; skills of carrying out physical and mathematical modeling, and also application of methods of the physical and mathematical analysis to the decision of concrete natural science and technical problems; \item to understand the logical connections between the sections of the course of physics, to develop the idea that physics is a universal basis for the technical Sciences, and that those physical phenomena and processes that are still limited in use in technology, in the future may be at the center of innovative achievements of engineering. \end{itemize}

Course objectives based on Bloom’s taxonomy

- What should a student remember at the end of the course?

By the end of the course, the students should be able to

• the basic physical phenomena and processes on which the principles of action of objects of professional activity, areas and possibilities of application of physical effects are based;
• fundamental concepts, laws and theories of classical and modern physics, limits of applicability of basic physical models;
• basic physical quantities and constants, their definitions and units of measurement;
• basic physical quantities and constants, their definitions and units of measurement;
• methods of physical research, including methods of modeling physical processes;
• methods for solving physical problems important for technical applications;
• physical bases of measurements, methods of measurement of physical quantities;
• technologies of work with different types of information;

- What should a student be able to understand at the end of the course?

By the end of the course, the students should be able to

• allocate physical content in systems and devices of different physical nature;
• carry out the correct mathematical description of physical phenomena in technological process;
• build and analyze mathematical models of physical phenomena and processes in solving applied problems;
• solve typical problems in the main branches of physics, using methods of mathematical analysis and modeling;
• apply concepts, physical laws and methods of problem solving to perform technical calculations, analysis and solution of practical problems, research in professional activities;
• to use modern physical equipment and devices in solving practical problems, to use the basic techniques of error estimation and experimental data processing;

- What should a student be able to apply at the end of the course?

By the end of the course, the students should be able to

• methods of analysis of physical phenomena in technical devices and systems;
• skills of practical application of the laws of physics, including in the design of products and processes;
• methods of theoretical research of physical phenomena and processes, construction of mathematical and physical models of real systems, solutions of physical problems;
• skills in the use of basic physical devices;
• methods of experimental physical research (planning, staging and processing of experimental data, including the use of standard software packages);
• skills of applying knowledge in the field of physics to study other disciplines.

Course evaluation

Course grade breakdown

Type	Points
Labs/seminar classes	0
Interim performance assessment	30
Exams	70

Grades range

Course grading range

Grade	Points
A	[85, 100]
B	[70, 84]
C	[50, 69]
D	[0, 49]

Resources and reference material

• Fundamentals of Physics (Halliday and Resnick) 10ed, ISBN 978-1-118-23072-5
• Arya A. Introduction to Classical Mechanics, Benjamin Cummings

Course Sections

The main sections of the course and approximate hour distribution between them is as follows:

Section 1

Section title

Kinematics of particles

Topics covered in this section

• Mathematical review (vectors)
• Measurements and One Dimension Motion (Along a Straight Line)
• Motion in Two and Three Dimensions

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. The position of a particle as it moves along a y axis is given by $y=2sin(\pi t/4)$, with t in seconds and y in centimeters. (a) What is the average velocity of the particle between t = 0 and t = 2.0 s? (b) What is the instantaneous velocity of the particle at t= 0, 1.0, and 2.0 s? (c) What is the average acceleration of the particle between t = 0 and t = 2.0 s? (d) What is the instantaneous acceleration of the particle at t = 0, 1.0, and 2.0 s?
2. A woman walks 250 m in the direction 30 $^{\circ }$ east of north, then 175 m directly east. Find (a) the magnitude and (b) the angle of her final displacement from the starting point. (c) Find the distance she walks. (d) Which is greater, that distance or the magnitude of her displacement?
3. Ship A is located 4.0 km north and 2.5 km east of ship B. Ship A has a velocity of 22 km/h toward the south, and ship B has a velocity of 40 km/h in a direction 37$^{\circ }$ north of east. (a) What is the velocity of A relative to B in unit-vector notation with toward the east? (b) Write an expression (in terms of and ) for the position of A relative to B as a function of t, where t=0 when the ships are in the positions described above. (c) At what time is the separation between the ships least? (d) What is that least separation?
4. A baseball is hit at Fenway Park in Boston at a point 0.762 m above home plate with an initial velocity of 33.53 m/s directed 55.0 $^{\circ }$ above the horizontal. The ball is observed to clear the 11.28-m-high wall in left field (known as the “green monster”) 5.00 s after it is hit, at a point just inside the left-field foul line pole. Find (a) the horizontal distance down the left-field foul line from home plate to the wall; (b) the vertical distance by which the ball clears the wall; (c) the horizontal and vertical displacements of the ball with respect to home plate 0.500 s before it clears the wall.

Typical questions for seminar classes (labs) within this section

1. Most important in an investigation of an airplane crash by the U.S. National Transportation Safety Board is the data stored on the airplane’s flight-data recorder, commonly called the “black box” in spite of its orange coloring and reflective tape.The recorder is engineered to withstand a crash with an average deceleration of magnitude 3400$g$ during a time interval of 6.50 ms. In such a crash, if the recorder and airplane have zero speed at the end of that time interval, what is their speed at the beginning of the interval?
2. Two vectors are given by \textbf{a}=3\textbf{i}+5\textbf{j} and \textbf{b}=2\textbf{i}+4\textbf{j}. Find (a) ${\textbf {a}}\times {\textbf {b}}$ (b) ${\textbf {a}}\cdot {\textbf {b}}$ (c) $({\textbf {a}}+{\textbf {b}})\cdot {\textbf {b}}$ (d) the component of \textbf{a} along the direction of \textbf{b}.
3. A cannon located at sea level fires a ball with initial speed 82 m/s and initial angle 45$^{\circ }$.The ball lands in the water after traveling a horizontal distance 686 m. How much greater would the horizontal distance have been had the cannon been 30 m higher?
4. An elevator without a ceiling is ascending with a constant speed of 10 m/s. A boy on the elevator shoots a ball directly upward, from a height of 2.0 m above the elevator floor, just as the elevator floor is 28 m above the ground.The initial speed of the ball with respect to the elevator is 20 m/s. (a) What maximum height above the ground does the ball reach? (b) How long does the ball take to return to the elevator floor?
5. A football player punts the football so that it will have a “hang time” (time of flight) of 4.5 s and land 46 m away. If the ball leaves the player’s foot 150 cm above the ground, what must be the (a) magnitude and (b) angle (relative to the horizontal) of the ball’s initial velocity?

Test questions for final assessment in this section

1. Two ships are moving parallel to each other in opposite directions with speeds $V_{1}$ and $V_{2}$. One ship shoots at the other. Find the angle of a gun to hit the target at the moment when distant between the ships are closest? The speed of the projectile $V_{0}$ is constant.
2. The velocity vector of a moving body is always parallel to acceleration vector. What is the trajectory of this body?
3. An object moves with non-constant velocity. Can the average velocity over a time interval $(t,t+T)$ be greater than or equal to the maximum instantaneous velocity at this time interval? Prove your answer.
4. A car starts moving with the initial zero velocity and with the acceleration, which depends on the time as Failed to parse (syntax error): {\displaystyle a(t) = 2(1 – exp(-t/15))} . Find the average velocity of the car over a time interval 10 s to 40 s.
5. A stone thrown at an angle $\alpha =30^{\circ }$ relative to the horizon has the same height H at moments $t_{1}=3$ s and $t_{2}=5$ s after start of his flying. Find the initial stone speed $v_{0}$ and height $H$.
6. A right angle is drawn on a paper. The ruler being always perpendicular to the bisector of this angle moves along this bisector at a speed of 10 cm/s. The ends of the ruler intersect the sides of the drawn angle. What is the velocity of the intersection points moving along the sides of the right angle relative to the paper?

Section 2

Section title

Kinetics of particles

Topics covered in this section

• Force and Motion
• Kinetic Energy and Work
• Potential Energy and Conservation of Energy

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. A vertical force \textbf{F} is applied to a block of mass m that lies on a floor.What happens to the magnitude of the normal force \textbf{$F_{N}$} on the block from the floor as magnitude F is increased from zero if force \textbf{F} is (a) downward and (b) upward?
2. A 1400 kg jet engine is fastened to the fuselage of a passenger jet by just three bolts (this is the usual practice). Assume that each bolt supports one-third of the load. (a) Calculate the force on each bolt as the plane waits in line for clearance to take off. (b) During flight, the plane encounters turbulence, which suddenly imparts an upward vertical acceleration of 2.6 $m/s^{2}$ to the plane. Calculate the force on each bolt now.
3. A person riding a Ferris wheel moves through positions at (1) the top, (2) the bottom, and (3) mid-height. If the wheel rotates at a constant rate, rank these three positions according to (a) the magnitude of the person’s centripetal acceleration, (b) the magnitude of the net centripetal force on the person, and (c) the magnitude of the normal force on the person, greatest first.
4. A box is on a ramp that is at angle $\theta$ to the horizontal. As $\theta$ is increased from zero, and before the box slips, do the following increase, decrease, or remain the same: (a) the component of the gravitational force on the box, along the ramp, (b) the magnitude of the static frictional force on the box from the ramp, (c) the component of the gravitational force on the box, perpendicular to the ramp, (d) the magnitude of the normal force on the box from the ramp, and (e) the maximum value $f_{s,max}$ of the static frictional force?
5. In three situations, a single force acts on a moving particle. Here are the velocities (at that instant) and the forces: (1) \textbf{v}=-4\textbf{i}, \textbf{F}=6\textbf{i}-20\textbf{j} (2) \textbf{v}=2\textbf{i}-3\textbf{j}, \textbf{F}=-2\textbf{i}+7\textbf{j} (3) \textbf{v}=-3\textbf{i}+1\textbf{j}, \textbf{F}=2\textbf{i}+6\textbf{j}. Rank the situations according to the rate at which energy is being transferred, greatest transfer to the particle ranked first, greatest transfer from the particle ranked last. \textbf{v}=-3\textbf{i}+1\textbf{j}, \textbf{F}=2\textbf{i}+6\textbf{j}.
6. What is the spring constant of a spring that stores 25 \textbf{J} of elastic potential energy when compressed by 7.5 cm?

Typical questions for seminar classes (labs) within this section

1. A shot putter launches a 7.260 kg shot by pushing it along a straight line of length 1.650 m and at an angle of 34.10$^{\circ }$ from the horizontal, accelerating the shot to the launch speed from its initial speed of 2.500 m/s (which is due to the athlete’s preliminary motion).The shot leaves the hand at a height of 2.110 m and at an angle of 34.10$^{\circ }$, and it lands at a horizontal distance of 15.90 m. What is the magnitude of the athlete’s average force on the shot during the acceleration phase? (Hint: Treat the motion during the acceleration phase as though it were along a ramp at the given angle.)
2. A 1000 kg boat is traveling at 90 km/h when its engine is shut off. The magnitude of the frictional force \textbf{$f_{k}$} between boat and water is proportional to the speed \textit{v} of the boat: $f_{k}=70v$, where \textit{v} is in meters per second and $f_{k}$ is in newtons. Find the time required for the boat to slow to 45 km/h.
3. A police officer in hot pursuit drives her car through a circular turn of radius 300 m with a constant speed of 80.0 km/h. Her mass is 55.0 kg. What are (a) the magnitude and (b) the angle (relative to vertical) of the net force of the officer on the car seat? (Hint: Consider both horizontal and vertical forces.)
4. A 0.250 kg block of cheese lies on the floor of a 900 kg elevator cab that is being pulled upward by a cable through distance $d_{1}$ = 2.40 m and then through distance $d_{2}$ = 10.5 m. (a) Through d1, if the normal force on the block from the floor has constant magnitude $F_{N}$ = 3.00 N, how much work is done on the cab by the force from the cable? (b) Through $d_{2}$, if the work done on the cab by the (constant) force from the cable is 92.61 kJ, what is the magnitude of $F_{N}$?
5. A block attached to a spring lies on a horizontal frictionless surface. The other end of the spring attached to the wall. The spring constant is 50 N/m. Initially, the spring is at its relaxed length and the block is stationary at position \textit{x} = 0. Then an applied force with a constant magnitude of 3.0 N pulls the block in the positive direction of the \textit{x} axis, stretching the spring until the block stops.When that stopping point is reached, what are (a) the position of the block, (b) the work that has been done on the block by the applied force, and (c) the work that has been done on the block by the spring force? During the block’s displacement, what are (d) the block’s position when its kinetic energy is maximum and (e) the value of that maximum kinetic energy?
6. A funny car accelerates from rest through a measured track distance in time \textit{T} with the engine operating at a constant power \textit{P}. If the track crew can increase the engine power by a differential amount \textit{dP}, what is the change in the time required for the run?
7. A spring with $k=100$ N/m is located at the top of a frictionless incline of angle 37 $^{\circ }$. The lower end of the incline is distance $D=$ 1.00 m from the end of the spring, which is at its relaxed length. A 2.00 kg canister is pushed against the spring until the spring is compressed 0.200 m and released from rest. (a) What is the speed of the canister at the instant the spring returns to its relaxed length (which is when the canister loses contact with the spring)? (b) What is the speed of the canister when it reaches the lower end of the incline?

Test questions for final assessment in this section

1. A slab of mass $m_{1}=$ 40 kg rests on a frictionless floor, and a block of mass $m_{2}=$ 10 kg rests on top of the slab. Between block and slab, the coefficient of static friction is 0.60, and the coefficient of kinetic friction is 0.40. A horizontal force \textbf{F} of magnitude 100 N begins to pull directly on the block, as shown. In unit-vector notation, what are the resulting accelerations of (a) the block and (b) the slab?
2. A box of canned goods slides down a ramp from street level into the basement of a grocery store with acceleration 0.75 $m/s^{2}$ directed down the ramp. The ramp makes an angle of 40$^{\circ }$ with the horizontal.What is the coefficient of kinetic friction between the box and the ramp?
3. A circular curve of highway is designed for traffic moving at 60 km/h. Assume the traffic consists of cars without negative lift. (a) If the radius of the curve is 150 m, what is the correct angle of banking of the road? (b) If the curve were not banked, what would be the minimum coefficient of friction between tires and road that would keep traffic from skidding out of the turn when traveling at 60 km/h?
4. An initially stationary 2.0 kg object accelerates horizontally and uniformly to a speed of 10 m/s in 3.0 s. (a) In that 3.0 s interval, how much work is done on the object by the force accelerating it? What is the instantaneous power due to that force (b) at the end of the interval and (c) at the end of the first half of the interval?
5. An iceboat is at rest on a frictionless frozen lake when a sudden wind exerts a constant force of 200 N, toward the east, on the boat. Due to the angle of the sail, the wind causes the boat to slide in a straight line for a distance of 8.0 m in a direction 20 $^{\circ }$ north of east. What is the kinetic energy of the iceboat at the end of that 8.0 m?
6. A boy is initially seated on the top of a hemispherical ice mound of radius $R$ = 13.8 m. He begins to slide down the ice, with a negligible initial speed. Approximate the ice as being frictionless. At what height does the boy lose contact with the ice?
7. The cable of the 1800 kg elevator cab snaps when the cab is at rest at the first floor, where the cab bottom is a distance $d$ = 3.7 m above a spring of spring constant $k$ = 0.15 MN/m. A safety device clamps the cab against guide rails so that a constant frictional force of 4.4 kN opposes the cab’s motion. (a) Find the speed of the cab just before it hits the spring. (b) Find the maximum distance $x$ that the spring is compressed (the frictional force still acts during this compression). (c) Find the distance that the cab will bounce back up the shaft. (d) Using conservation of energy, find the approximate total distance that the cab will move before coming to rest. (Assume that the frictional force on the cab is negligible when the cab is stationary.)

Section 3

Section title

Kinetics of systems of particles

Topics covered in this section

• Center of Mass and Linear Momentum
• Rotation
• Rolling, Torque, and Angular Momentum

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. Ricardo, of mass 80 kg, and Carmelita, who is lighter, are enjoying Lake Merced at dusk in a 30 kg canoe.When the canoe is at rest in the placid water, they exchange seats, which are 3.0 m apart and symmetrically located with respect to the canoe’s center. If the canoe moves 40 cm horizontally relative to a pier post, what is Carmelita’s mass?
2. A steel ball of mass 0.500 kg is fastened to a cord that is 70.0 cm long and fixed at the far end.The ball is then released when the cord is horizontal. At the bottom of its path, the ball strikes a 2.50 kg steel block initially at rest on a frictionless surface. The collision is elastic. Find (a) the speed of the ball and (b) the speed of the block, both just after the collision.
3. A tall, cylindrical chimney falls over when its base is ruptured. Treat the chimney as a thin rod of length 55.0 m.At the instant it makes an angle of 35.0$^{\circ }$ with the vertical as it falls, what are (a) the radial acceleration of the top, and (b) the tangential acceleration of the top. (Hint: Use energy considerations, not a torque.) (c) At what angle u is the tangential acceleration equal to g?
4. A cannonball and a marble roll smoothly from rest down an incline. Is the cannonball’s (a) time to the bottom and (b) translational kinetic energy at the bottom more than, less than, or the same as the marble’s?
5. A solid brass cylinder and a solid wood cylinder have the same radius and mass (the wood cylinder is longer). Released together from rest, they roll down an incline. (a) Which cylinder reaches the bottom first, or do they tie? (b) The wood cylinder is then shortened to match the length of the brass cylinder, and the brass cylinder is drilled out along its long (central) axis to match the mass of the wood cylinder.Which cylinder now wins the race, or do they tie?

Typical questions for seminar classes (labs) within this section

1. Particle $A$ and particle $B$ are held together with a compressed spring between them.When they are released, the spring pushes them apart, and they then fly off in opposite directions, free of the spring.The mass of $A$ is 2.00 times the mass of $B$, and the energy stored in the spring was 60 J. Assume that the spring has negligible mass and that all its stored energy is transferred to the particles. Once that transfer is complete, what are the kinetic energies of (a) particle $A$ and (b) particle $B$?
2. Block 2 (mass 1.0 kg) is at rest on a frictionless surface and touching the end of an unstretched spring of spring constant 200 N/m.The other end of the spring is fixed to a wall. Block 1 (mass 2.0 kg), traveling at speed $v_{1}$ = 4.0 m/s, collides with block 2, and the two blocks stick together.When the blocks momentarily stop, by what distance is the spring compressed?
3. A 1400 kg car moving at 5.3 m/s is initially traveling north along the positive direction of a $y$ axis. After completing a 90 right-hand turn in 4.6 s, the inattentive operator drives into a tree, which stops the car in 350 ms. In unit-vector notation, what is the impulse on the car (a) due to the turn and (b) due to the collision? What is the magnitude of the average force that acts on the car (c) during the turn and (d) during the collision? (e) What is the direction of the average force during the turn?
4. A pulsar is a rapidly rotating neutron star that emits a radio beam the way a lighthouse emits a light beam.We receive a radio pulse for each rotation of the star.The period T of rotation is found by measuring the time between pulses.The pulsar in the Crab nebula has a period of rotation of $T$ = 0.033 s that is increasing at the rate of $1.26\times 10^{-5}s/y$. (a) What is the pulsar’s angular acceleration a? (b) If a is constant, how many years from now will the pulsar stop rotating? (c) The pulsar originated in a supernova explosion seen in the year 1054.Assuming constant a, find the initial $T$.
5. A pulley, with a rotational inertia of $1.0\times 10^{-3}$ $kgm^{2}$ about its axle and a radius of 10 cm,is acted on by a force applied tangentially at its rim.The force magnitude varies in time as $F=0.50t+0.30t^{2}$, with $F$ in newtons and $t$ in seconds.The pulley is initially at rest.At $t$=3.0 s what are its (a) angular acceleration and (b) angular speed?
6. bowling ball of radius $R$ = 11 cm along a lane. The ball slides on the lane with initial speed $v_{com,0}$ = 8.5 m/s and initial angular speed $\omega _{0}=0$. The coefficient of kinetic friction between the ball and the lane is 0.21.The kinetic frictional force \textbf{$f_{k}$} acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular acceleration of the ball. When speed $v_{c}om$ has decreased enough and angular speed $\omega$ has increased enough, the ball stops sliding and then rolls smoothly. (a) What then is $v_{c}om$ in terms of $\omega$? During the sliding, what are the ball’s (b) linear acceleration and (c) angular acceleration? (d) How long does the ball slide? (e) How far does the ball slide? (f) What is the linear speed of the ball when smooth rolling begins?

Test questions for final assessment in this section

1. A thin solid disc rolls without slipping on the surface of a hemispherical pit. What is depth where the disc pressure on the pit wall is equal to its weight? The radius of the pit $R$ is much larger than the radius of the disc $r$.
2. Firefighters sometimes use a high-pressure fire hose to knock down the door of a burning building. Suppose such a hose delivers 22 kg of water per second at a velocity of 16 m/s. Assuming the water hits and runs straight down to the ground (that is, it doesn't bounce back), what average force is exerted on the door?
3. A rigid hoop of radius $R$ and mass $M$ is lying on a horizontal frictionless table and pivoted at the point $P$. A point-like object of mass m with the velocity $V_{i}$ collides elastically with the resting hoop. After collision the point-like object moves with an unknown velocity $V_{f}$ in an opposite direction with respect to its initial motion. Find the linear velocity $V_{f}$ of the point-like object and the angular velocity $\omega _{f}$ of the hoop after collision.
4. A homogeneous elastic rod of mass $M$ lies on a smooth horizontal table. An elastic ball of mass m hits the end of the rod, moving at a velocity $v$ perpendicular to the rod. Find the ball velocity at the moment when the deformation energy of the rod and the ball is maximal. The friction between the rod and the table should be neglected; the inertia moment of the rod related to the center of the mass is $I=mL^{2}/12$, where $L$ is length of the rod.