BSc: Physics I

Physics I (mechanics)

• Course name: Physics I (mechanics)
• Code discipline: XYZ
• Subject area:

Short Description

This course covers the following concepts: fundamental concepts of physics for calculating problems of mechanics in statics, dynamics..

Prerequisites

Prerequisite subjects

• CSE201 — Mathematical Analysis I: functions, limits, derivatives, definite and indefinite integrals, exponentials.
• CSE203 MathematicalAnalysis II

Prerequisite topics

• CSE204 Geometry And Linear Algebra II: vector and matrix operations, spatial analysis (unit vectors, rotations)
• CSE205 Differential Equations: first- and second-order ODEs

Course Topics

Course Sections and Topics

Section	Topics within the section
Kinematics of particles	Mathematical review (vectors) Measurements and One Dimension Motion (Along a Straight Line) Motion in Two and Three Dimensions
Kinetics of particles	Force and Motion Kinetic Energy and Work Potential Energy and Conservation of Energy
Kinetics of systems of particles	Center of Mass and Linear Momentum Rotation Rolling, Torque, and Angular Momentum
Statics	Equilibrium
Oscillations	Harmonic motion

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

to study physical phenomena and fundamental laws of physics, the limits of their applicability,, to get the students acquainted with the basic mechanical quantities, to know their definitions, meaning, methods and units of their measurement,, to highlight the most fundamental physical experiments and their role in the development of science as a whole,, to acquire skills of carrying out physical and mathematical modeling and obtain thee skills of physical and mathematical analysis on the example of concrete natural science and technical problems,, to understand 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 find limited application nowadays may become the center of innovative achievements of engineering in the future.

ILOs defined at three levels

Level 1: What concepts should a student know/remember/explain?

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

• basic physical quantities and constants, their definitions and units of measurement,
• physical bases of measurements, methods of measurement of physical quantities;
• fundamental concepts, laws and theories of classical mechanics, limits of applicability of basic physical models,
• basic concepts and methods of solving kinematic problems,
• peculiarities of describing static and dynamic mechanical systems,
• how to apply the laws of conservation of energy and momentum when modeling dynamical mechanical systems,
• methods of modeling mechanical motion,
• methods for solving physical problems in real applications,
• vocabulary representing a neutral scientific style, as well as the basic terminology of mechanics and physics.

Level 2: What basic practical skills should a student be able to perform?

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

• handle different physical quantities based on their dimensionality,
• use basic kinematical formulas to calculate the motion of solid bodies,
• apply Newton's laws to analyze the motion of bodies and mechanical systems under the effect of external and internal forces,
• calculate kinetic and potential energy of mechanical systems,
• use the laws of conservation of energy and momentum when modeling the dynamics of mechanical systems,
• calculate physical characteristics of rotating extended bodies (angular velocities, accelerations, moment of inertia, momentum, torque),
• use the laws of conservation of momentum in calculations of systems with extended rotating bodies,
• independently collect, systematize, analyze and competently use information from theoretical sources, including reference books and standards.

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

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

• the methods of analysis of mechanical 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.

Grading

Course grading range

Grade	Range	Description of performance
A. Excellent	85-100	-
B. Good	70-84	-
C. Satisfactory	50-69	-
D. Poor	0-49	-

Course activities and grading breakdown

Activity Type	Percentage of the overall course grade
Quizzes	50
Homework assignments	20
Labs/seminar classes	5
Final exam	25
Attendance	5

Resources, literature and reference materials

Open access resources

• Halliday, Resnick, and Walker, 2018. Fundamentals of Physics (11th Ed.) John Wiley & Sons, , ISBN 978-1-119-28624-0
• Arya A. Introduction to Classical Mechanics.

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Activities within each section

Learning Activities	Section 1	Section 2	Section 3	Section 4	Section 5
Homework and group projects	1	1	1	1	1
Midterm evaluation	1	1	1	1	1
Testing (written or computer based)	1	1	1	1	1
Discussions	1	1	1	1	1

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type	Content	Is Graded?
Question	The position of a particle as it moves along a y axis is given by ${\textstyle y=2sin(\pi t/4)}$ , 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?	1
Question	A woman walks 250 m in the direction 30 ${\textstyle ^{\circ }}$ 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?	1
Question	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${\textstyle ^{\circ }}$ 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?	1
Question	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 ${\textstyle ^{\circ }}$ 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.	1
Question	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${\textstyle g}$ 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?	0
Question	Two vectors are given by a=3i+5j and b=2i+4j. Find (a) ${\textstyle {\textbf {a}}\times {\textbf {b}}}$ (b) ${\textstyle {\textbf {a}}\cdot {\textbf {b}}}$ (c) ${\textstyle ({\textbf {a}}+{\textbf {b}})\cdot {\textbf {b}}}$ (d) the component of a along the direction of b.	0
Question	A cannon located at sea level fires a ball with initial speed 82 m/s and initial angle 45${\textstyle ^{\circ }}$ .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?	0
Question	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?	0
Question	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?	0

Section 2

Activity Type	Content	Is Graded?
Question	A vertical force F is applied to a block of mass m that lies on a floor.What happens to the magnitude of the normal force ${\textstyle F_{N}}$ on the block from the floor as magnitude F is increased from zero if force F is (a) downward and (b) upward?	1
Question	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 ${\textstyle m/s^{2}}$ to the plane. Calculate the force on each bolt now.	1
Question	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.	1
Question	A box is on a ramp that is at angle ${\textstyle \theta }$ to the horizontal. As ${\textstyle \theta }$ 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 ${\textstyle f_{s,max}}$ of the static frictional force?	1
Question	In three situations, a single force acts on a moving particle. Here are the velocities (at that instant) and the forces: (1) v=-4i, F=6i-20j (2) v=2i-3j, F=-2i+7j (3) v=-3i+1j, F=2i+6j. 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. v=-3i+1j, F=2i+6j.	1
Question	What is the spring constant of a spring that stores 25 J of elastic potential energy when compressed by 7.5 cm?	1
Question	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${\textstyle ^{\circ }}$ 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${\textstyle ^{\circ }}$ , 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.)	0
Question	A 1000 kg boat is traveling at 90 km/h when its engine is shut off. The magnitude of the frictional force ${\textstyle f_{k}}$ between boat and water is proportional to the speed v of the boat: ${\textstyle f_{k}=70v}$ , where v is in meters per second and ${\textstyle f_{k}}$ is in newtons. Find the time required for the boat to slow to 45 km/h.	0
Question	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.)	0
Question	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 ${\textstyle d_{1}}$ = 2.40 m and then through distance ${\textstyle d_{2}}$ = 10.5 m. (a) Through d1, if the normal force on the block from the floor has constant magnitude ${\textstyle F_{N}}$ = 3.00 N, how much work is done on the cab by the force from the cable? (b) Through ${\textstyle d_{2}}$ , if the work done on the cab by the (constant) force from the cable is 92.61 kJ, what is the magnitude of ${\textstyle F_{N}}$ ?	0
Question	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 x = 0. Then an applied force with a constant magnitude of 3.0 N pulls the block in the positive direction of the 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?	0
Question	A funny car accelerates from rest through a measured track distance in time T with the engine operating at a constant power P. If the track crew can increase the engine power by a differential amount dP, what is the change in the time required for the run?	0
Question	A spring with ${\textstyle k=100}$ N/m is located at the top of a frictionless incline of angle 37 ${\textstyle ^{\circ }}$ . The lower end of the incline is distance ${\textstyle D=}$ 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?	0

Section 3

Activity Type	Content	Is Graded?
Question	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?	1
Question	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.	1
Question	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${\textstyle ^{\circ }}$ 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?	1
Question	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?	1
Question	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?	1
Question	Particle ${\textstyle A}$ and particle ${\textstyle B}$ 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 ${\textstyle A}$ is 2.00 times the mass of ${\textstyle B}$ , 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 ${\textstyle A}$ and (b) particle ${\textstyle B}$ ?	0
Question	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 ${\textstyle v_{1}}$ = 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?	0
Question	A 1400 kg car moving at 5.3 m/s is initially traveling north along the positive direction of a ${\textstyle y}$ 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?	0
Question	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 ${\textstyle T}$ = 0.033 s that is increasing at the rate of ${\textstyle 1.26\times 10^{-5}s/y}$ . (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 ${\textstyle T}$ .	0
Question	A pulley, with a rotational inertia of ${\textstyle 1.0\times 10^{-3}}$ ${\textstyle kgm^{2}}$ 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 ${\textstyle F=0.50t+0.30t^{2}}$ , with ${\textstyle F}$ in newtons and ${\textstyle t}$ in seconds.The pulley is initially at rest.At ${\textstyle t}$ =3.0 s what are its (a) angular acceleration and (b) angular speed?	0
Question	bowling ball of radius ${\textstyle R}$ = 11 cm along a lane. The ball slides on the lane with initial speed ${\textstyle v_{com,0}}$ = 8.5 m/s and initial angular speed ${\textstyle \omega _{0}=0}$ . The coefficient of kinetic friction between the ball and the lane is 0.21.The kinetic frictional force ${\textstyle f_{k}}$ 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 ${\textstyle v_{c}om}$ has decreased enough and angular speed ${\textstyle \omega }$ has increased enough, the ball stops sliding and then rolls smoothly. (a) What then is ${\textstyle v_{c}om}$ in terms of ${\textstyle \omega }$ ? 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?	0

Section 4

Activity Type	Content	Is Graded?
Question	A ladder leans against a frictionless wall but is prevented from falling because of friction between it and the ground. Suppose you shift the base of the ladder toward the wall. Determine whether the following become larger, smaller, or stay the same (in magnitude): (a) the normal force on the ladder from the ground, (b) the force on the ladder from the wall, (c) the static frictional force on the ladder from the ground, and (d) the maximum value ${\textstyle f_{s,max}}$ of the static frictional force.	1
Question	An automobile with a mass of 1360 kg has 3.05 m between the front and rear axles. Its center of gravity is located 1.78 m behind the front axle.With the automobile on level ground, determine the magnitude of the force from the ground on (a) each front wheel (assuming equal forces on the front wheels) and (b) each rear wheel (assuming equal forces on the rear wheels).	1
Question	A uniform cubical crate is 0.750 m on each side and weighs 500 N. It rests on a floor with one edge against a very small, fixed obstruction. At what least height above the floor must a horizontal force of magnitude 350 N be applied to the crate to tip it?	1
Question	A trap door in a ceiling is 0.91 m square, has a mass of 11 kg, and is hinged along one side, with a catch at the opposite side. If the center of gravity of the door is 10 cm toward the hinged side from the door’s center, what are the magnitudes of the forces exerted by the door on (a) the catch and (b) the hinge?	1
Question	A door has a height of 2.1 m along a y axis that extends vertically upward and a width of 0.91 m along an ${\textstyle x}$ axis that extends outward from the hinged edge of the door.A hinge 0.30 m from the top and a hinge 0.30 m from the bottom each support half the door’s mass, which is 27 kg. In unit-vector notation, what are the forces door at (a) the top hinge and (b) the bottom hinge?	0
Question	A cubical box is filled with sand and weighs 890 N.We wish to “roll” the box by pushing horizontally on one of the upper edges. (a) What minimum force is required? (b) What minimum coefficient of static friction between box and floor is required? (c) If there is a more efficient way to roll the box, find the smallest possible force that would have to be applied directly to the box to roll it. (Hint: At the onset of tipping, where is the normal force located?)	0
Question	A crate, in the form of a cube with edge lengths of 1.2 m, contains a piece of machinery; the center of mass of the crate and its contents is located 0.30 m above the crate’s geometrical center.The crate rests on a ramp that makes an angle ${\textstyle \theta }$ with the horizontal.As ${\textstyle \theta }$ is increased from zero, an angle will be reached at which the crate will either tip over or start to slide down the ramp. If the coefficient of static friction ${\textstyle \mu _{s}=0.7}$ between ramp and crate is 0.60, (a) does the crate tip or slide and (b) at what angle u does this occur? If ${\textstyle \mu _{s}=0.7}$ , (c) does the crate tip or slide and (d) at what angle ${\textstyle \theta }$ does this occur? (Hint: At the onset of tipping, where is the normal force located?)	0
Question	A beam of length L is carried by three men, one man at one end and the other two supporting the beam between them on a crosspiece placed so that the load of the beam is equally divided among the three men. How far from the beam’s free end is the crosspiece placed? (Neglect the mass of the crosspiece.)	0
Question	The leaning Tower of Pisa is 59.1 m high and 7.44 m in diameter. The top of the tower is displaced 4.01 m from the vertical. Treat the tower as a uniform, circular cylinder. (a) What additional displacement, measured at the top, would bring the tower to the verge of toppling? (b) What angle would the tower then make with the vertical?	0

Section 5

Activity Type	Content	Is Graded?
Question	What is the maximum acceleration of a platform that oscillates at amplitude 2.20 cm and frequency 6.60 Hz?	1
Question	Two particles oscillate in simple harmonic motion along a common straight-line segment of length A. Each particle has a period of 1.5 s, but they differ in phase by ${\textstyle \pi }$ rad. (a) How far apart are they (in terms of A) 0.50 s after the lagging particle leaves one end of the path? (b) Are they then moving in the same direction, toward each other, or away from each other?	1
Question	A thin uniform rod (mass = 0.50 kg) swings about an axis that passes through one end of the rod and is perpendicular to the plane of the swing. The rod swings with a period of 1.5 s and an angular amplitude of 10${\textstyle ^{\circ }}$ . (a) What is the length of the rod? (b) What is the maximum kinetic energy of the rod as it swings?	1
Question	A 1000 kg car carrying four 82 kg people travels over a “washboard” dirt road with corrugations 4.0 m apart. The car bounces with maximum amplitude when its speed is 16 km/h. When the car stops, and the people get out, by how much does the car body rise on its suspension?	1
Question	A massless spring hangs from the ceiling with a small object attached to its lower end.The object is initially held at rest in a position ${\textstyle y_{i}}$ such that the spring is at its rest length. The object is then released from ${\textstyle y_{i}}$ and oscillates up and down, with its lowest position being 10 cm below ${\textstyle y_{i}}$ . (a) What is the frequency of the oscillation? (b) What is the speed of the object when it is 8.0 cm below the initial position? (c) An object of mass 300 g is attached to the first object, after which the system oscillates with half the original frequency. What is the mass of the first object? (d) How far below ${\textstyle y_{i}}$ is the new equilibrium (rest) position with both objects attached to the spring?	0
Question	The suspension system of a 2000 kg automobile “sags” 10 cm when the chassis is placed on it. Also, the oscillation amplitude decreases by 50 % each cycle. Estimate the values of (a) the spring constant k and (b) the damping constant b for the spring and shock absorber system of one wheel, assuming each wheel supports 500 kg.	0
Question	The scale of a spring balance that reads from 0 to 15.0 kg is 12.0 cm long. A package suspended from the balance is found to oscillate vertically with a frequency of 2.00 Hz. (a) What is the spring constant? (b) How much does the package weigh?	0
Question	When a 20 N can is hung from the bottom of a vertical spring, it causes the spring to stretch 20 cm. (a) What is the spring constant? (b) This spring is now placed horizontally on a frictionless table. One end of it is held fixed, and the other end is attached to a 5.0 N can.The can is then moved (stretching the spring) and released from rest.What is the period of the resulting oscillation?	0

Final assessment

Section 1

1. Two ships are moving parallel to each other in opposite directions with speeds ${\textstyle V_{1}}$ and ${\textstyle V_{2}}$ . 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 ${\textstyle V_{0}}$ 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 ${\textstyle (t,t+T)}$ 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): {\textstyle a(t) = 2(1 – exp(-t/15))}

. Find the average velocity of the car over a time interval 10 s to 40 s.

1. A stone thrown at an angle ${\textstyle \alpha =30^{\circ }}$ relative to the horizon has the same height H at moments ${\textstyle t_{1}=3}$ s and ${\textstyle t_{2}=5}$ s after start of his flying. Find the initial stone speed ${\textstyle v_{0}}$ and height ${\textstyle H}$ .
2. 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

1. A slab of mass ${\textstyle m_{1}=}$ 40 kg rests on a frictionless floor, and a block of mass ${\textstyle m_{2}=}$ 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 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 ${\textstyle m/s^{2}}$ directed down the ramp. The ramp makes an angle of 40${\textstyle ^{\circ }}$ 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 ${\textstyle ^{\circ }}$ 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 ${\textstyle R}$ = 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 ${\textstyle d}$ = 3.7 m above a spring of spring constant ${\textstyle k}$ = 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 ${\textstyle x}$ 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

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 ${\textstyle R}$ is much larger than the radius of the disc ${\textstyle r}$ .
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 ${\textstyle R}$ and mass ${\textstyle M}$ is lying on a horizontal frictionless table and pivoted at the point ${\textstyle P}$ . A point-like object of mass m with the velocity ${\textstyle V_{i}}$ collides elastically with the resting hoop. After collision the point-like object moves with an unknown velocity ${\textstyle V_{f}}$ in an opposite direction with respect to its initial motion. Find the linear velocity ${\textstyle V_{f}}$ of the point-like object and the angular velocity ${\textstyle \omega _{f}}$ of the hoop after collision.
4. A homogeneous elastic rod of mass ${\textstyle M}$ lies on a smooth horizontal table. An elastic ball of mass m hits the end of the rod, moving at a velocity ${\textstyle v}$ 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 ${\textstyle I=mL^{2}/12}$ , where ${\textstyle L}$ is length of the rod.

Section 4

1. A uniform ladder whose length is 5.0 m and whose weight is 400 N leans against a frictionless vertical wall. The coefficient of static friction between the level ground and the foot of the ladder is 0.46.What is the greatest distance the foot of the ladder can be placed from the base of the wall without the ladder immediately slipping?
2. A 73 kg man stands on a level bridge of length ${\textstyle L}$ . He is at distance ${\textstyle L/4}$ from one end. The bridge is uniform and weighs 2.7 kN.What are the magnitudes of the vertical forces on the bridge from its supports at (a) the end farther from him and (b) the nearer end?
3. A uniform cube of side length 8.0 cm rests on a horizontal floor.The coefficient of static friction between cube and floor is ${\textstyle \mu }$ . A horizontal pull P is applied perpendicular to one of the vertical faces of the cube, at a distance 7.0 cm above the floor on the vertical midline of the cube face. The magnitude of P is gradually increased. During that increase, for what values of ${\textstyle \mu }$ will the cube eventually (a) begin to slide and (b) begin to tip? (Hint: At the onset of tipping, where is the normal force located?)
4. A pan balance is made up of a rigid, massless rod with a hanging pan attached at each end. The rod is supported at and free to rotate about a point not at its center. It is balanced by unequal masses placed in the two pans.When an unknown mass ${\textstyle m}$ is placed in the left pan, it is balanced by a mass ${\textstyle m_{1}}$ placed in the right pan; when the mass ${\textstyle m}$ is placed in the right pan, it is balanced by a mass ${\textstyle m_{2}}$ in the left pan. Show that ${\textstyle m={\sqrt {m_{1}m_{2}}}}$

Section 5

1. pendulum is formed by pivoting a long thin rod about a point on the rod. In a series of experiments, the period is measured as a function of the distance ${\textstyle x}$ between the pivot point and the rod’s center. (a) If the rod’s length is ${\textstyle L}$ = 2.20 m and its mass is ${\textstyle m}$ = 22.1 g, what is the minimum period? (b) If ${\textstyle x}$ is chosen to minimize the period and then ${\textstyle L}$ is increased, does the period increase, decrease, or remain the same? (c) If, instead,${\textstyle m}$ is increased without ${\textstyle L}$ increasing, does the period increase, decrease, or remain the same?
2. A block weighing 10.0 N is attached to the lower end of a vertical spring (${\textstyle k}$ = 200.0 N/m), the other end of which is attached to a ceiling.The block oscillates vertically and has a kinetic energy of 2.00 J as it passes through the point at which the spring is unstretched. (a) What is the period of the oscillation? (b) Use the law of conservation of energy to determine the maximum distance the block moves both above and below the point at which the spring is unstretched. (These are not necessarily the same.) (c) What is the amplitude of the oscillation? (d) What is the maximum kinetic energy of the block as it oscillates?
3. A simple harmonic oscillator consists of an 0.80 kg block attached to a spring (${\textstyle k}$ = 200 N/m). The block slides on a horizontal frictionless surface about the equilibrium point ${\textstyle x}$ = 0 with a total mechanical energy of 4.0 J. (a) What is the amplitude of the oscillation? (b) How many oscillations does the block complete in 10 s? (c) What is the maximum kinetic energy attained by the block? (d) What is the speed of the block at ${\textstyle x}$ = 0.15 m?
4. A damped harmonic oscillator consists of a block (${\textstyle m}$ = 2.00 kg), a spring (${\textstyle k}$ = 10.0 N/m), and a damping force (${\textstyle F=-bv}$ ). Initially, it oscillates with an amplitude of 25.0 cm; because of the damping, the amplitude falls to three-fourths of this initial value at the completion of four oscillations. (a) What is the value of ${\textstyle b}$ ? (b) How much energy has been “lost” during these four oscillations?

The retake exam

Section 1

Section 2

Section 3

Section 4

Section 5