Honors Physics Chapter 5 Forces in Two Dimensions

Honors Physics Turn in H4 & W4 Lecture Q&A

Vector and Scalar Vector: Scalar: Magnitude: How large, how fast, … Direction: In what direction (moving or pointing) Representation depends on frame of reference Position, displacement, velocity, acceleration, force, momentum, … Scalar: Magnitude only No direction Representation does not depend on frame of reference Mass, distance, length, speed, energy, temperature, charge, …

Vector symbol Vector: bold or an arrow on top Scalar: regular Typed: v and V or Handwritten: Scalar: regular v or V v stands for the magnitude of vector v.

Graphical representation of vector: Arrow An arrow is used to graphically represent a vector. The direction of the arrow represents the direction of the vector. The length of the arrow represents the magnitude of the vector. (Vector a is smaller than vector b because a is shorter than b.) head b a tail When comparing the magnitudes of vectors, we ignore directions.

Equivalent Vectors Two vectors are identical and equivalent if they both have the same magnitude and are in the same direction. A B C A, B and C are all equivalent vectors. They do not have to start from the same point. (Their tails don’t have to be at the same point.)

Negative of Vector Vector -A has the same magnitude as vector A but points in the opposite direction. If vector A and B have the same magnitude but point in opposite directions, then A = -B, and B = -A A -A -A

Adding Vectors Graphical Algebraic (by components) Head-to-Tail (Triangular) Algebraic (by components)

Adding Vectors: Head-to-Tail Head-to-Tail method: B A Example: A + B Draw vector A Draw vector B starting from the head of A A+B B The vector drawn from the tail of A to the head of B is the sum of A + B. A Make sure arrows are parallel and of same length.

A+B=B+A B A A + B A B+A B How about B + A? What can we conclude? A+B B

A+B+C B A C C B A+B+C A Resultant vector: from tail of first to head of last.

A-B=A+(-B) A B A A-B -B Draw vector A Draw vector -B from head of A. The vector drawn from the tail of A to head of –B is then A – B.

Think … Two forces are acting on an object simultaneously, one is 4 N and the other is 5 N. a) What is the maximum possible resultant force on the object? How are these two forces oriented relative to each other when this happens? 4N 5N 9 N, same direction 9N b) What is the minimum possible resultant force on the object? How are these two forces oriented relative to each other when this happens? 4N 1 N, opposite direction 5N 1N

What are the relations among the vectors? B B A + C C A + C = B A c a + b b a b c a + b = -c a + b + c = 0 a

Using ruler and protractor, we find: Example: Vector a has a magnitude of 5.0 units and is directed east. Vector b is directed 35o west of north and has a magnitude of 4.0 units. Construct vector diagrams for calculating a + b and b – a. Estimate the magnitudes and directions of a + b and b – a from your diagram. E N W S 35o -a b b-a a+b 35o 66o 50o a Using ruler and protractor, we find: a+b: 4.3 unit, 50o North of East b-a: 8.0 unit, 66o West of North

Law of cosine b a c C a b c C = 90o Pythagorean’s Theorem:

Law of Sine or a c b B A C

Example: 125-7 You first walk 8 Example: 125-7 You first walk 8.0 km north from home, then walk east until your displacement from home is 10.0 km. How far east did you walk? E N b a r

Practice: 121-3 A hiker walks 4 Practice: 121-3 A hiker walks 4.5 km in one direction, then makes an 45o turn to the right and walks another 6.4 km. What is the magnitude of her displacement? a c 135o 45o b

Vector Components y a ay x ax Drop perpendicular lines from the head of vector a to the coordinate axes, the components of vector a can be found: y a ay  x ax  is the angle between the vector and the +x axis. ax and ay are scalars.

Useful Trigonometry opposite side hypotenuse  adjacent side

Finding components of a vector Resolving the vector Decomposing the vector ax is the component of a in the x-direction the x component of a. Component form:

Practice: A heavy box is pulled across a wooden floor with a rope Practice: A heavy box is pulled across a wooden floor with a rope. The rope makes an angle of 60o with the floor. A force of 75 N is exerted on the rope. What are the components of the force parallel and perpendicular to the floor? Ty T  Tx

Vector magnitude and direction y ay a Vector magnitude and direction  ax x The magnitude and direction of a vector can be found if the components (ax and ay) are given: (for 3-D)  is the angle from the +x axis to the vector. Magnitude and direction form:

Example: A car is driven 125. 0 km due west, then 65. 0 km due south Example: A car is driven 125.0 km due west, then 65.0 km due south. What is the magnitude and direction of its displacement? Set up the frame of reference as to the right. Then y a x b r  Magnitude is 141 km, at 27.5o South of West.

Adding Vectors by Components x y ry r by b Adding Vectors by Components ay bx a rx ax When adding vectors by components, we add components in a direction separately from other components. 2-D 3-D

Let x = horizontal, y = vertical, then Practice: 125-8 A child’s swing is held up by two ropes tied to a tree branch that hangs 13.0o from the vertical. If the tension in each rope is 2.28 N, what is the combined force (magnitude and direction) of the two ropes on the swing? a+b b b a W Let x = horizontal, y = vertical, then

Friction Friction: force opposing the motion or tendency of motion between two rough surfaces that are in contact Static friction is not constant and has a maximum: Kinetic (or sliding) friction is constant: N: normal force between the two surfaces s (and k) is coefficient of static (and kinetic) friction.  depends on the properties of the two surfaces  = 0 when one of the surfaces is smooth (frictionless.) s > k for same surfaces.  has no unit.

Example: What happens to the frictional force as you increase the force pushing on a table on the floor? Fapp f But what if the applied force increases just slightly? f Table not moving yet. Static friction increases as applied force increases. fs = Fapp sN • Once table is moving, a smaller constant kinetic friction. fk < fs, max • • • • Fapp sN

Example: 128-17 A girl exerts a 36-N horizontal force as she pulls a 52-N sled across a cement sidewalk at constant speed. What is the coefficient of kinetic friction between the sidewalk and the metal sled runners? Ignore air resistance. + We only consider the motion in the horizontal direction. N= 52N Fnet needs to include only forces in the horizontal direction. Define right to be the positive direction. f Fp=36N W=52N

Practice: 128-18 You need to move a 105-kg sofa to a different location in the room. It takes a force of 102 N to start it moving. What is the coefficient of static friction between the sofa and the carpet? + Let right = + direction. Fnet includes only forces in the horizontal direction. Then N f Fp=102 N W

Equilibrium and Equilibrant Force Concurrent forces: forces acting on the same object at the same time. Equilibrium: Fnet = 0 Equilibrant Force: A force that produce equilibrium when applied to an object. C B C is the equilibrant force of A + B because (A + B) + C = 0. A A + B + C = C = - (A + B)

Example Find the equilibrant force c to a + b if a and b are given as followed. What is the relationship between c and a + b? a + b c b a b a c = - (a + b) a + b + c = 0

Example: Two forces act on an object Example: Two forces act on an object. A 36-N force acts at 225o, a 48-N force acts at 315o. What would be the magnitude and direction of their equilibrant? x y FC FA FA+FB FB

More Application on Force: Example: 135-40 Stacie, who has a mass of 45 kg, starts down a slide that is inclined at an angle of 45o with the horizontal. If the coefficient of kinetic friction between Stacie’s shorts and the slide is 0.25, what is her acceleration? x: y y: N x f Wx W Wy What if frictionless incline?

Practice: What is the acceleration of a box on a smooth (frictionless) incline that makes an angle of 30o with the horizontal, as in the diagram? x: y y: N x Wx W Wy

Weight hanging What are the two tensions if the block has a mass of 5 Weight hanging What are the two tensions if the block has a mass of 5.0 kg and the upper string makes an angle of 30o with the horizontal? y Hanging mass: T2 T2y x • P T2x N T1 Consider forces acting on point P: y: T1 + 5.0kg W