PHYS 201 STUDY GUIDE FOR PART TWO:

FORCES AND MOTION

Dr. Johnny B. Holmes

## Introduction

In this second part of the course we consider: (a) two dimensional motion [combining 1-D motion with vectors]; (b) the relation between force and motion [which introduces the concept of mass and Newton's three laws of motion]; and (c) one of the four fundamental forces: gravity.

We begin this section by looking at two dimensional motion. This combines the ideas of position, velocity, and acceleration with the idea of a vector (having components). We look at three special cases: trajectory motion, uniform circular motion, and relative velocity.

We next 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: S F = m a, called NEWTON'S SECOND LAW. We also consider Newton's first (which is a special case of Newton's Second Law, and Newton's third law which is important in determining who is pushing on whom.

As an important example of force, we consider Newton's Law of Gravity and how it relates to the concept of GRAVITATIONAL MASS.

Next we consider the concept of torque (or angular force) and its relation to rotations and force.

## Two Dimensional Motion

1. Projectile motion (work in components!) M
1. x(t) and y(t)
2. Uniform circular motion (turning but not speeding up) N
1. frequency
1. f is frequency in cycles/sec
2. w = 2p f, w is angular velocity in radians/sec
3. T = 1/f, T is period in sec/cycle
2. tangential velocity: vtang = w r = 2p r/T (circular means r=constant so vradial = 0)
3. radial acceleration: aradial = w vtang = w ²r = vtang²/r ;
4. tangential acceleration causes speed to change, so uniform circular motion means atang = 0
3. Relative velocity (addition of vectors!) O

M. A ball is thrown from the top of a 14 m high building with a speed of 25 m/s directed at an angle of 33° above the horizontal. a) How far away will the ball hit the ground? b) How long will it take the ball before it hits the ground?

N. The earth moves with (approximately) uniform circular motion about its axis. The earth's radius = 6,378 kilometers. a) What is the period of the earth's rotation about its axis? b) What is the tangential speed of the earth at the equator as it spins? c) What is the angular velocity of the earth as it spins? d) Is the earth's surface undergoing acceleration as it spins? e) What is the radial acceleration at the equator of the earth in it spins? f) What is the tangential acceleration of the earth at the equator as it spins?

O. A boat can travel at a speed of 5 m/sec on a calm lake. a) What direction sould 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?

M. a) 75.01 m; b) 3.58 sec.

N. a) T = 1 day = 86,400 sec; b) 463 m/s = 1,039 mph;

c) 7.27 x 10-5 rad/s; d) YES; e) 0.034 m/s² ; f) 0 m/s² .

O. a) 23.6° upstream; b) 43.6 sec.; c) straight across; d) 40 sec.

## The Laws of Motion

1. The relation of force and motion: Newton's Laws of Motion P
1. inertial mass: quality of matter
2. Newton's second law: SF = ma [i.e. SFx = max , SFy = may , SFz = maz ]
1. units of mass (kg)
2. units of force (Nt)
3. this is a vector equation - COMPONENTS!!!
3. Newton's first and third laws
1. first law: special case (SF=0) of the second law
2. third law: action-reaction, who pushes on whom, can't push yourself
2. Applications Q,R,S
1. force of gravity (on surface of earth) Fgr = Weight = m g, down
2. force of contact (or normal force): Fc is balancing, PERPENDICULAR to surface
3. friction: Ffr £ m Fc ( i.e., balancing up to a point), PARALLEL to surface
4. spring force Fspring = -k x
5. x and y component equations
6. parallel and perpendicular component equations
7. circular motion and centripetal force (Fcent is due to real forces, it is not a different force!)
8. pulleys and mechanical advantage (count the ropes attached)

3. Gravitational force T
1. gravitational mass: quality of matter: Fgravity µ m
2. gravitational mass & inertial mass: different qualities, same quantity
3. Newton's law of gravity: F = Gm1m2/r² where G=6.67x10-11 Nt m²/kg²
4. mass and weight: Weight = Fgravity = m g, where g = G M /
5. gravity, weight, and g
1. g on earth: g = G Mearth/Rearth² = 9.8 m/s²
2. g above earth: g = G Mearth / (Rearth + h)²; gabove < gsurface
3. g on other planets, moons, asteroids, etc.
4. Orbits: Circular motion due to gravity U
1. Fgr = m acircular, or GMearthmsat/r² = msat w²r
5. Artificial weight: Circular motion seems like gravity V
1. geffective = acircular = w ²r
6. Torque and rotations W,X,Y
1. torque causes rotations
2. t = r F sin(q ) where q is the angle between r and F
3. statics: S F=0 and S t = 0
4. mechanical advantage: levers and clubs

LETTER PROBLEMS:

P. A body hangs from a spring balance supported from the roof of an elevator. (a) If the elevator has an upward acceleration of 2 m/s² and the balance reads 200 Nt., what is the true weight of the body? (b) Under what circumstances will the balance read 150 Nt? (c) What will the balance read if the elevator cable breaks?

Q. Given the force as a function of time: F(t) = 5 Nt cos[(0.3 rad/sec)t]. a) graph the force versus time; b) graph the acceleration versus time; c) graph the velocity versus time; d) graph the position versus time. R. What is the acceleration of block A in the figure to the right given that the mass of A is 5 kg, the mass of B is 3 kg, and the coefficient of friction between A and the table is 0.15 ?

S. Given that an object of mass 3 kg starts from the origin (xo =0) with a speed of 5 m/s (vo =5m/s), and given that it is attached to a spring that exerts a force Fs(x) = -kx where k = 1.2 Nt/m [the spring constant], 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 second. *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. See what happens to the motion as your vary the mass.

T. What is the acceleration due to gravity on Mars? (Mars has a mass of 6.4 x 1023 kg and a radius of 3400 km.

U. a) At what radius should a geosynchronous satellite (period = 24 hours) of mass 50 kg orbit the earth? b) What speed should this satellite have to remain in orbit? c) What is the gravitational force on the satellite in orbit? d) How does this compare to the gravitational force on the satellite when it was on the earth's surface getting ready for launch?

V. 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?

W. Consider a car of known mass, m, going around a turn of radius, r, as fast as it can without sliding off the road (don't worry about flipping over here). Consider that the curve is banked at some angle, q, with respect to the horizontal, and that the tire-road coefficient of friction is m. a) Draw a picture of the situation, and draw in arrows indicating all the forces that act on the car during the turn. (You do not have to consider the force due to the engine or the force due to air resistance.) b) Draw an arrow indicating the direction of the acceleration during the turn. c) Set up Newton's Equations of Motion for this situation. d) Indicate what quantities are knwon and what are unknown in the problem. e) Indicate any other equations that can be used to solve the problem.

X. a) Estimate the distance from your elbow to where your biceps are connected to your forearm. b) Estimate the distance your hand is from your elbow. c) Neglecting the weight of the forearm and hand, how much force must the biceps exert for you to hold a 2 lb object in your hand (assume that the forearm is horizontal and the elbow is bent at an angle of 90°) ?

Y. You are to pick up and hold horizontal a board that has a weight of 25 lb. and a length of 8 ft. You do so by lifting up 1 ft from the left end with the right hand and pushing down on the left end (at the end) with the left hand. a) What is the force that the right hand must exert to hold the board horizontal? b) What is the force that the left hand must exert?

P. (a) m = 16.95 kg; W = 166.1 Nt; (b) a = -.95 m/s² ; (c) T = 0 Nt.

Q. 2.76 m/s² .

R. You are on your own.

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

T. .38 * gearth = 3.7 m/s².

U. (a) 42,300 km; (b) 3,076 m/s; (c) 11.2 Nt; (d) 11.2/490, or 2%.

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

W. You are on your own.