# Vector & Scalar Quantities

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1 Chapter 2 Forces and Vectors Vector & Scalar Quantities Vector Quantities Vectors are physical quantities that have both magnitude and direction. Magnitude = amount and units. Direction can be stated as up/down, left/right, N/E/S/W or 35 o S of E. Eg. of vectors: displacement, velocity, acceleration, force, and momentum.

2 Vectors are sometimes represented by a line and arrow drawn on the line. The length of the line represents magnitude of the vector quantity. Arrow on the line represents direction. When asked to specify a vector quantity, state both its magnitude (size and units) as well as its direction. More about Vectors in Chapter 4!! Scalar Quantities: Scalar quantities are physical quantities that have only magnitude. Scalars do not require direction in space when specifying them. Eg: distance, speed, mass, time, temperature and energy.

3 2.1: Forces The physical universe is made of objects (particles) that interact with each other. The interaction may define or change the behavior (temperature, motion) of the interacting objects. Effects of these interactions are explained in different ways (models) such as force, momentum exchange, energy, etc. We will first use force as a means of understanding some of these interactions. Force: = Push or pull one object exerts on another. Forces come in pairs, isolated forces do not exist in physical interactions. Eg. When you push the door, the door pushes back on you.

4 The SI unit of force is the Newton (N). 1N = 1 kg-m/s 2. In the US, force is measured in pounds lb = N and 1.00 N = lb. Robert Hooke ( ) found that when a spring is pulled by a force F, it extends proportionally by an amount x. Hooke s law: F = -kx k = spring constant, indicates how stiff the spring is. Measuring Forces Weight = force of gravity pulling on objects. Weight is measured using calibrated spring scale.

5 Examples of Forces: Contact forces eg. Applied force of push/pull, force of tension in strings, friction, normal force, spring force. Long-range, action-at-a-distance (noncontact) forces. Eg. Gravitational (between earth-moon), Electric, magnetic. Weak nuclear force. Strong nuclear force. All forces fall under 4 fundamental categories: Gravitational (always attractive). Electromagnetic (all contact forces). Strong Nuclear (holds protons and neutrons together. Weak Nuclear (occurs in some forms of radioactivity and thermonuclear reactions in the sun).

6 Newton s Laws of Motion Newton s First Law (Also called Law of Inertia) An object at rest will remain at rest and an object in motion will continue to move with constant speed and direction unless acted upon by a net force. Other ways of stating it: If no net force acts on an object at rest, it will remain at rest but if the object is already moving, it will continue to move without change in its speed and/or its direction. If the sum of all forces acting on an object is zero, then its speed and direction will not change.

7 net force = vector sum of all forces. A net force is needed to make an object at rest start moving. A net force is needed to make a moving object change direction of motion. A net force is needed to stop a moving object. A force is not needed to keep an object in motion if there is no force opposing its motion. Inertia Inertia = resistance to change in motion. Mass = amount of inertia of an object. A larger mass has more resistance to change in its motion than a smaller mass. An object at rest wants to stay at rest, an object in motion along a straight line wants to keep moving that way unless acted on by a net force. [inertia = resistance to change in motion] Newton s Law one Law of inertia. Seatbelts are worn because of inertia. Newton s first law closely related to the reason why seatbelts are worn by motorist.

8 A force of 15 N is applied to the end of a spring, and it stretches 9 cm. How much farther will it stretch if an additional 5.0 N of force is applied? (A) 3.0 cm (B)1.67 cm (C)10.67 cm (D)15 cm (E) 5.0 cm If the net force acting on a moving object suddenly becomes zero, the object will (A) continue moving but with non zero acceleration. (B) stop abruptly. (C) continue moving at constant velocity. (D) slow down gradually.

9 Velocity (v) Velocity (v) in simple terms is speed and direction. (Better definition later). If a nonzero net force (F net ) is applied to an object, its velocity will change. Thus, if the net force acting on an object is zero, its velocity will not change, i.e. no change in its speed, no change in its direction. Net force (resultant force) = vector sum of all the forces acting on an object. Acceleration (a) Change in velocity gives rise to acceleration (a). If an object moves with changing velocity, we say the object moves with an acceleration. Acceleration is rate of change of velocity. ie a = v/ t ( means change ). Change in velocity could mean (a) change in speed only - while direction stays constant. (b)change in direction only while speed stays constant. (c) change in speed and direction of motion simultaneously. F net = ma

10 Newton s Second Law: The greater the net force, the greater the acceleration, ie, a F net The greater the mass of the object, the less acceleration, ie, a 1/m. The direction of the acceleration is the same as the direction of the net force. Thus, acceleration a F net /m or a = F net /m. From Newton s second law of motion, we have the relation F net = ma

11 Static and Dynamic Equilibrium If the net force on an object is zero, it could either: Be at rest static equilibrium. Be moving with zero acceleration ie no change in velocity constant velocity dynamic (or translational) equilibrium. If the net force acting on an object is not zero, the object will move with changing velocity (acceleration). Newton s Third Law of Motion In an interaction between two objects, the forces that each exerts on the other are equal in magnitude but opposite in direction. To every action, there is an equal an opposite reaction

12 Note that the two equal and opposite forces are not acting on the same object!! Action/Reaction Scenarios: A person throws a package out of a boat at rest. Boat starts to move in opposite direction. Ice skater pushes against railing and moves in opposite direction. Rocket exerts strong force expelling gases. Gases exerts equal force in opposite direction, propelling the rocket forward. 2.4: Net Force and Vector Addition Net force = vector sum of all the forces acting on an object. Vectors are added in a special way. Co-linear vectors 2 or more vectors parallel or antiparallel.

13 + y - + x - Force Laws

14 1. Gravitational Forces: Newton s law of universal gravitation states that any two objects of masses m 1 and m 2 separated by a distance r will exert a gravitational force on each other. This gravitational force is attractive force and is directly proportional to the product of the masses (F m 1 m 2 ) and inversely proportional to r 2 (F 1/r 2 ). F m 1 m 2 and F 1/r 2 combine to give F = Gm 1 m 2 /r 2 G = Universal Gravitational constant = x N.m 2 /kg 2 Objects near the surface of the earth, gravitational force is called weight, W. An object of mass m near the surface of the earth has weight W = mg g = acceleration due to gravity = 9.8 m/s 2.

15 W = mg = GM E m/r E2 or g = GM E /R 2 E Acceleration due to gravity, g, is directed downwards, towards the center of the earth. Far away from the surface of the earth, (r = R E + h), the magnitude of g (and therefore the weight of an object at that location), decreases: g = GM E /r 2 = GM E /(R E + h) 2 Weight (W) Is the force of gravity due to the pull of the earth. g = Acceleration due to gravity = 9.8 m/s 2. Hence for an object of mass m, the weight is W = mg Direction of W is always straight downward - ie. Toward the center of the earth.

16 2. Spring Force Spring or elastic string stretched or compressed by distance x. The force that restores the spring (string) to its x original length is given by the expression F = -kx x [Hooke s Law]. Negative sign is because direction of F is always opposite to the direction of x. 3. Normal Force (N) Consider a book of mass m at rest on a table. N By Newton s law, since the book is at rest, the net force on it must be zero. Hence the table must be exerting an mg upward force on the book to cancel out the force of gravity. In this case, N = mg. [It is not always that N = mg!!] Normal force is a contact force and is the force on an object when it is in contact with a surface. It is always directed perpendicularly away from the surface, ie normally.

17 4. Friction Friction is a contact force between an object and a surface, and directed parallel to the surface. There are two types of friction: (a) Static Friction: (f s ) Is the frictional force that exists when there is no sliding or skidding between an object and a surface. Increases to a maximum value f s (max) when the object starts to slide against the surface. 0 f s f fmax Maximum static friction f f (max) = µ s N µ s = coefficient of static friction. (b) Kinetic (sliding) Friction: Is the frictional force that exists when an object slides against a surface. f k = µ k N, where µ k is coefficient of kinetic friction. Usually, µ k µ s so static friction > kinetic friction.

18 Free Body Diagram A sketch drawing to help find net force acting on an isolated (free) body. Draw the object. May be represented by just a dot. Draw all forces acting on the object. The length of the line and arrows should represent the forces as closely as possible. Do not include forces acting on other objects. The net force is obtained by performing vector addition of all the forces drawn. Identify all forces acting on: 1. A wooden block sliding down an incline plane. 2. A wooden block sliding up an incline plane. 3. One of the tires of a car skidding on a flat road. 4. One of the tires of a car moving normally on a flat road. 5. A stone in mid air going upward. 6. A stone in mid air coming downward.

19 Object A is moving with constant velocity. Object B is at rest. What does A and B have in common? (A) Acceleration not zero but constant. (B) Acceleration is zero. (C) A non-zero net force acts on them. (D) Same mass and weight. The forces acting on a plane are: Lift L = 14 kn up, Weight W = -14 kn down, Thrust T = 0.8 kn east, and Drag D = 1.2 kn west. What is the net force acting on it?

20 The moon: Radius = 1.74 x 10 6 m Mass = 7.35 x kg What would be the magnitude of g acting on a mass m placed near the moon s surface? F = Gm 1 m 2 /r 2, Weight W = mg x N.m 2 /kg 2 A force of 10 N is applied to the end of a spring, and it stretches 5 cm. How much further will it stretch if an additional 5 N of force is applied? Hooke s law: F = -kx

21 A box of weight 50 N is at rest on a floor where µ s = 0.3. A rope is attached to the box and pulled horizontally with tension T = 30 N. Will the box move?

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