Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 3: Acceleration and Newton’s Second Law of Motion Position & Displacement Speed & Velocity Acceleration Newton’s Second Law Relative Velocity

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 §3.1 Position & Displacement The position (r) of an object describes its location relative to some origin or other reference point. The displacement is the change in an object’s position. It depends only on the beginning and ending positions.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3 Example (text problem 3.4): Margaret walks to the store using the following path: miles west, miles north, miles east. What is her total displacement? Give the magnitude and direction. x y r3r3 r2r2 r1r1 rr Take north to be in the +y direction and east to be along +x.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 Example continued: The displacement is  r = r f – r i. The initial position is the origin; what is r f ? The final position will be r f = r 1 + r 2 + r 3. The components are r fx = -r 1 + r 3 = -0.2 miles and r fy = +r 2 = +0.2 miles. Using the figure, the magnitude and direction of the displacement are x y rr ryry rxrx  N of W.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 §3.2 Velocity Velocity is a vector that measures how fast and in what direction something moves. Speed is the magnitude of the velocity. It is a scalar.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 Path of a particle Start finish rr v av is the constant speed that results in the same displacement in a given time interval.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 y x r0r0 rfrf Points in the direction of  r rr v0v0 The instantaneous velocity points tangent to the path. vfvf A particle moves along the blue path as shown. At time t 1 its position is r 0 and at time t 2 its position is r f.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 On a graph of position versus time, the average velocity is represented by the slope of a chord. x (m) t (sec) t1t1 t2t2 x1x1 x2x2

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 This is represented by the slope of a line tangent to the curve on the graph of an object’s position versus time. x (m) t (sec)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 The area under a velocity versus time graph (between the curve and the time axis) gives the displacement in a given interval of time. v(m/s) t (sec)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 Example: Consider Margaret’s walk to the store in the example on slides 3 and 4. If the first leg of her walk takes 10 minutes, the second takes 8 minutes, and the third 7 minutes, compute her average velocity and average speed during each leg and for the overall trip. Use the definitions:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12 Leg  t (hours) v av (miles/hour) Average speed (miles/hour) (west) (north) (east)2.56 Total trip (45  N of W) 2.40 Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 Example (text problem 3.20): Speedometer readings are obtained and graphed as a car comes to a stop along a straight-line path. How far does the car move between t = 0 and t =16 seconds? Since there is not a reversal of direction, the area between the curve and the time axis will represent the distance traveled.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 Example continued: The rectangular portion has an area of Lw = (20 m/s)(4 s) = 80 m. The triangular portion has an area of ½bh = ½(8 s) (20 m/s) = 80 m. Thus, the total area is 160 m. This is the distance traveled by the car.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 §3.3 Newton’s Second Law of Motion A nonzero acceleration changes an object’s state of motion. These have interpretations similar to v av and v.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 y x v0v0 r0r0 rfrf vfvf A particle moves along the blue path as shown. At time t 1 its position is r 0 and at time t 2 its position is r f. vv Points in the direction of  v. The instantaneous acceleration can point in any direction.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 Example (text problem 3.31): If a car traveling at 28 m/s is brought to a full stop 4.0 s after the brakes are applied, find the average acceleration during braking. Given: v i = +28 m/s, v f = 0 m/s, and  t = 4.0 s.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 Example (text problem 3.39): At the beginning of a 3 hour trip you are traveling due north at 192 km/hour. At the end, you are traveling 240 km/hour at 45  west of north. (a) Draw the initial and final velocity vectors. x (east) y (north) v0v0 vfvf

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 (b) Find  v. The components are South of west Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 (c) What is a av during the trip? The magnitude and direction are: South of west Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 Newton’s 2nd Law: The acceleration of a body is directly proportional to the net force acting on the body and inversely proportional to the body’s mass. Mathematically:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 An object’s mass is a measure of its inertia. The more mass, the more force is required to obtain a given acceleration. The net force is just the vector sum of all of the forces acting on the body, often written as  F.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23 If a = 0, then  F = 0. This body can have: Speed = 0 which is called static equilibrium, or speed  0, but constant, which is called dynamic equilibrium.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 §3.4 Applying Newton’s Second Law Force units: 1 N = 1 kg m/s 2.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 Example: Find the tension in the cord connecting the two blocks as shown. A force of 10.0 N is applied to the right on block 1. Assume a frictionless surface. The masses are m 1 = 3.00 kg and m 2 = 1.00 kg. F block 2block 1 Assume that the rope stays taut so that both blocks have the same acceleration.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 FBD for block 2: T F w1w1 N1N1 x y x T w2w2 N2N2 y FBD for block 1: Apply Newton’s 2 nd Law to each block:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 These two equations contain the unknowns: a and T. To solve for T, a must be eliminated. Solve for a in (2) and substitute in (1). (1) (2) Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 Example: A box slides across a rough surface. If the coefficient of kinetic friction is 0.3, what is the acceleration of the box? FkFk w N x y FBD for box: Apply Newton’s 2 nd Law:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29 (1) (2) From (1): Solving for a: Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 §3.5 Relative Velocity Example: You are traveling in a car (A) at 60 miles/hour east on a long straight road. The car (B) next to you is traveling at 65 miles/hour east. What is the speed of car B relative to car A?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31 From the picture: A B A B t=0 t>0  r AG  r BG  r BA Divide by  t: +x Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 32 Example: You are traveling in a car (A) at 60 miles/hour east on a long straight road. The car (B) next to you is traveling at 65 miles/hour west. What is the speed of car B relative to car A?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33 A B A B t=0  r AG  r BG  r BA t>0 +x From the picture: Divide by  t: Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 34 Summary Position Displacement Versus Distance Velocity Versus Speed Acceleration Instantaneous Values Versus Average Values Newton’s Second Law Relative Velocity

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 35 A ball incident on a wall has a speed of 10 m/s toward the wall. It rebounds with a speed of 10 m/s. What is the direction of the ball’s acceleration while it is in contact with the wall? A.Toward the wall B.Away from the wall C.Up the wall D.Down the wall E.The ball is not accelerated. Additional clicker question: