PHYS 150

Dr. Johnny B. Holmes

Introduction
Motion in a Plane
The Laws of Motion
Gravity
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In this second quarter of the course we extend our consideration of motion to include: (a) motion in two dimensions; (b) the relation between force and motion; (c) a fundamental quality of matter: mass, and its connection to motion; and (d) several kinds of force including one of the four fundamental forces: gravity.

We first consider motion in a plane. To do this we combine the concept of vector developed in the first Part with the concepts of motion also developed in the first part.

Next we introduce the concept of inertial mass as a basic quality of matter. It is the quality that determines how a force will affect the motion of an object. This extremely important relation between force and motion is contained in a very simple equation: (SF = ma), called Newton's Second Law. We also consider Newton's first and third laws of motion in this part.

In this part we also consider the concept of gravitational mass and how it relates to the fundamental gravitational force.

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Outline
Supplementary Homework Problems
Answers to Supplementary Problems
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1. Projectile motion (work in components!)
2. Uniform circular motion
   [uniform means speed=constant; circular means radius=constant]
1. frequency
1. f is frequency in cycles/second, or cps
2. w =2pf, where w is angular velocity in radians/second
3. T = 1/f, where T is the period in seconds/cycle
2. tangential velocity: v_t = wr (radial velocity = 0 since r = constant)
3. radial acceleration: a_r = wv = w²r = v²/r [since direction changes]
   tangential acceleration: a_t = 0 [since speed does not change].
3. Relative velocity

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13. A person throws a ball from the top of a building 12 meters high with an initial speed of 35 m/s at an angle of 62 degrees above the horizontal. Assume no air resistance. a) How high does the ball go? b) How far from the building does the ball hit the ground?

14. The earth goes around the sun in one year. The earth-sun distance is 93 million miles (149 million kilometers). a) How fast is the earth moving around the sun? b) What is the angular velocity, w, of the earth around the sun? c) What is the magnitude and direction of the acceleration of the earth around the sun?

15. A boat can travel at a speed of 5 m/sec on a calm lake. a) What direction should a pilot point his boat to go directly across a river that flows at a rate of 2 m/sec? b) How long will it take if the river is 200 meters wide? c) What direction should he head his boat if he wants to just get across the river in the least time? d) how long will be the least time? e) What direction should a pilot point his boat to reach a point 20 meters upstream? f) How long will this trip take?

16. (extra credit) The water in a river is 300 meters wide and flows at a uniform rate of 6 m/sec. The boat (same as in problem 15 above with speed of 5 m/s) cannot go straight across the river so the man must walk back upstream if he wishes to reach a point directly across from his starting point. The man can walk at a rate of 2 m/sec. a) What angle should he point his boat to get to the point directly opposite in the least amount of time? b) How long will it take?

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13. a) 60.72 meters (at 3.153 sec.); b) 109.66 meters (at 6.674 sec.).

14. a) 29,700 m/s (66,500 mph); b) 1.99x10^-7 rad/sec; c) 0.0059 m/s² toward the sun.

15. a) 23.6^o upstream; b) 43.6 sec.; c) straight across; d) 40 sec. e&f) you are on your own.

16. a) 38.7^o upstream; b) 187.3 sec.

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Outline
Supplementary Homework Problems
Answers to Supplementary Homework Problems
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1. The relation of force and motion
1. inertial mass: a quality a matter
2. Newton's Second Law: SF=ma, or SF_x=ma_x, SF_y=ma_y
1. units of mass (kg)
2. units of force (Nt)
3. this is a vector equation - work in rectangular components
3. Newton's first and third laws
1. first law: special case of the second law
2. third law: action-reaction, or can't push yourself
2. Basic Forces
1. weight is a force: magnitude: weight = W = mg, where g = 9.8 m/s², directed down;
2. contact force: magnitude: balancing; directed perpendicular to surface;
3. friction force: magnitude: balancing up to a point: F_{f £} mF_c, directed parallel to surface;
4. tension: magnitude: same at both ends; directed parallel to string direction;
5. spring: magnitude: k(x-x_o); directed towards the equilibrium point (x_o)
3. Applications
1. x and y component equations
2. parallel and perpendicular component equations

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17. A 4 kg block is accelerated upward by a cord whose breaking strength is 100 Nt. Find the maximum acceleration which can be given the block without breaking the cord.

18. An object hangs from a spring balance supported from the roof of an elevator. a) If the elevator has an upward acceleration of 1.5 m/s², and the balance reads 750 Nts., what is the true weight of the body? b) Under what circumstances will the balance read 600 Nts? c) What will the balance read if the cable breaks?

19. Be able to draw a picture of the motion of a projectile (projected at some initial velocity at 45^o above horizontal) under the following situations: a) no gravity, no air resistance; b) constant gravity with no air resistance; c) constant gravity with heavy air resistance.

20. Be able to draw a picture of the motion of a ball initially on a string going around a circle under the following situations: a) no gravity, no resistance, the string breaks; b) on a horizontal table with vertical gravity, no friction, no air resistance, the string breaks; c) with gravity, no air resistance, circle is in vertical plane, string breaks on upward part of path; d) with gravity, with air resistance, circle is in vertical plane, string breaks on upward part of path.

21. Given: the object starts from rest [v_xo0] at the origin [x_o=0] and has a mass of 3 kg, and the only force is: F_x(t) = 5 Nt sin[(0.3 rad/sec)t] is applied from t=0 to t=10 sec. [Warning: the argument of the cosine function is in radians, not degrees!] a) Find the acceleration at: (1) t=0 sec, (2) t=1 sec, and (3) t=5 sec. b) Find by numerical methods (i.e., assume the acceleration is constant during the first second, etc) the velocity at: (1) t=1 sec, (2) t=2 sec, and (3) t=5 sec. [Warning: to do this you will need to find the acceleration for t=3 and t=4 seconds also.] c) Find by numerical methods the approximate position of the object at t=5 sec. Note warning above! d) Find the velocity as a function of time, v_x(t) , using the calculus; compare the value from this function at t=5 seconds to your approximate answer in part b. e) Find the position as a function of time, x(t), using calculus; compare the value from this function at t=5 seconds to your answer in part c.

22. Given the force as a function of time (as in problem 21 above): a) graph the force versus time; b) graph the acceleration versus time; c) graph the velocity versus time; d) graph the position versus time.

23. What is the acceleration of block A, currently at rest on a horizontal table with a string attached that extends over a pulley and is connected to block B which is hanging? Block A has a mass of 5 kg, block B has a mass of 4 kg, and the coefficient of friction between A and the table is 0.35.

24. A car of mass 2,000 kg has an engine that we will assume can put out a constant force of 4000 Nt (not a realistic assumption). In all cases we will assume the car starts from rest at time zero. a) Assuming no air resistance or other frictions, what will the velocity of the car be: (1) after 5 seconds, (2) after 100 seconds, and (3) after 200 seconds? b) (Use a programmable calculator or computer and show your program as well as the answer.) Assuming an air resistance can be described by F_{air resistance} = bv², where b = 0.4 Nt-s²/m², and assuming no other frictions, use numerical methods to find out what the velocity of the car will be: (1) after 5 seconds, (2) after 100 seconds, and (3) after 200 seconds. [Be sure in the look back stage to see if you understand the behavior with and without the air resistance.]

25. Given that an object of mass 3 kg starts from the origin (x_o=0) with a speed of 5 m/s (v_xo=5 m/s), and given that it is attached to a spring that exerts a force: F_x-spring = -kx, where k = 24 Nt/m [the spring constant], and given that no other forces act on the object, find: a) the acceleration at t=0 when the object starts; b) assuming that the object experiences this acceleration for 0.1 seconds, find [by numerical methods] its approximate new speed and new position at t=0.1 sec; c) do this process nine more times to find the approximate speed and position at the end of the first full second; d) graph the acceleration versus time, the velocity versus time, and the position versus time, all between t=0 and t=1 seconds. *e) [extra credit] Try using a computer program (or program your calculator) to do the above numerical approximation, and follow the motion for several seconds.

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17. 15.2 m/s² .

18. a) m = 66.37 kg; W = 650.4 Nt; b) a = -0.76 m/s²; c) T = 0 Nt.

19 & 20. The drawing should show the slope of the line at the initial break point, whether it ever reaches a highest point, whether it ever falls straight down or otherwise travels in a straight line.

21 & 22. You are on your own - the answers to different parts should agree.

23. 2.45 m/s².

24. a-1) v = 10 m/s, a-2) v = 200 m/s; a-3) v = 400 m/s; b-1,2,3) you are on your own.

25. You are on your own. See if your answer corresponds to what you would expect of a spring.

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Outline
Supplementary Homework Problems
Answers to Problems
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1. Gravitational Force
1. gravitational mass: quality of matter
2. gravitational mass and inertial mass: same value but different properties
3. Newton's law of gravity: F_gravity = Gm₁m₂/r² , attractive; where G = 6.67x10^-11 Nt-m²/kg².
4. gravity, weight and g:
1. F_gravity = Weight = mg, where g = GM/r²
2. g on earth's surface: g = GM_earth/R_earth² = 9.8 m/s²
3. g above earth's surface: r > r_Rearth , so g_above < g_surface
4. g on other planets and the moon
2. Artificial weight and accelerated frames

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SUPPLEMENTARY HOMEWORK PROBLEMS (S-):

26. What is the acceleration due to gravity on Mars? Mars has a mass of 6.4 x 10²³ kg [so M_Mars = 0.11 M_earth] and a radius of 3,400 km [R_Mars = 0.53 R_earth].

27. Two identical, isolated particles, each of mass 2 kg, are separated by a distance of 30 cm. what is the magnitude of the gravitational force of one particle on the other?

28. A satellite is to be put into a geosynchronous orbit (T=24 hours). a) At what height should the satellite orbit? b) At what speed should the satellite orbit?

29. A space station is built in the shape of a cylinder of length 1 km and radius 200 m, and is spun about the axis of the cylinder. a) If an effective gravity of 60% that of earth is desired, what should the period of the spin be? b) How fast is the outside edge moving? c) Which direction is up?

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26. 0.38 gearth = 3.7 m/s² .

27. 2.96 x 10^-9 Nt.

28. a) r = R_earth + h = 42,000 km; b) v = 3,000 m/s = 6,700 mph.

29. a) 36.64 sec; b) 34.29 m/s; c) toward the axis.

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