PHYSICS
General Corrections: 1. Fix indentations 2. Bold terminologies and important concepts
CLASSIFICATION OF QUANTITIES A. Scalar quantity – has only magnitude and is completely specified by a number and a unit. Ex: distance, speed, work (work is a vector quantity since W=Fd and F is a vector) more examples: mass, time B. Vector quantity – has both magnitude and direction and it can be represented by an arrow . The length of the arrow represents the magnitude and the direction and the arrowhead point represents the direction of the vector. Symbols of vector quantities are printed in boldface type or expressed in handwriting by arrows over the letters. Ex: displacement, velocity, force more example: acceleration Vector Addition A. Graphical Method 2. 1. Parallelogram *insert picture* 1. 2. Polygon/Triangle (head-to-tail) *insert picture* B. Trigonometric Method 1. If two vectors (A and B) are perpendicular to each other:
B
R θ
A
Magnitude of the resultant is given by the Pythagorean Theorem: θ
Angle ₱ between R and A (direction):
2. If two vectors are not perpendicular, use Sine and/or Cosine Law: Sine Law: Cosine Law:
C. Component Method Rules on Component Method 1. Resolve the initial vectors into components in the x and y directions. 2. Add the components in the x direction to give Rx and add the components in the y direction to give Ry. Rx = Ax + Bx + Cx + … Ry = Ay + By + Cy + … 3. Calculate the magnitude and direction of the resultant R from its components Rx and Ry. Magnitude:
Direction:
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Sample Problem (Vector Addition): 120° Two tugboats are towing a ship. Each exerts a force of 600 N, and the angle between two ropes is 600. What is the resultant force on the ship? 60
A
600N
Ship
120°
B
600N
Solution: A. Using Trigonometric Method To add the force vectors A and B, vector B is shifted parallel to itself so that its tail is at the head of A. To get the length of the resultant R
A 30deg
= 600N 120°
B = 600N 30deg
R
B . Using Component Method The angle between the forces each tugboat exerts on the ship and the direction of the ship’s motion is 300. A
30o
B
R
30o
(Trigo Method)
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same as above
R= , O 0° Product of Vectors: The dot product (or also called “scalar product”) A · B = AB cos φ = ABA Where: A and B = magnitudes of vectors A and B, respectively Φ = angle between vectors A and B The cross product or also called “vector product” C = A x B = (AB sin φ) n Where: A and B = magnitudes of vectors A and B respectively ₱ = angle between vectors A and B (0°<₱<180°) n = unit vector perpendicular to both A and B not important for UPCAT
A x B =
Right-hand rule:
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Problem-solving: “I SEE” strategy
t t t t
Identify the relevant concept Set up the problem Execute the solution Evaluate your answer
distorts flow, move to Tips section
.MECHANICS Two Parts of Mechanics A. Statics – analysis of bodies at rest B. Dynamics – deals with the analysis of bodies in motion. a. Kinematics – analysis of the geometry of motion. Kinematics is used to relate displacement, velocity, acceleration, and time, without reference to the cause of motion. b. Kinetics – study of the relation existing between the forces acting on a body, the mass of the body, and the motion of the body. TERMS USED TO DESCRIBE POSITION AND MOTION Distance – total length traveled Displacement – change in position with respect to the original position Speed – refers to how far an object travels on a given time period Velocity – signifies how fast an object is moving and the direction in which it moves; rate at which position changes Average Speed – defined as the distance traveled divided by the time it takes to travel this distance Average Velocity – defined as the quotient of the displacement Δx and the time interval Δt
Instantaneous Velocity – refers to the velocity of an object at an instant; average velocity of an object over an indefinitely short time interval
Acceleration – refers to the rate at which velocity changes; change of velocity divided by the time taken to make the change. Average Acceleration – defined as the change in velocity divided by the change in time
Instantaneous Acceleration – refers to the acceleration of an object at an instant.
Uniform Rectilinear Motion – the acceleration of the particle is zero for every value of time. The velocity is therefore constant.
where
x = final position x0 = initial position v = velocity (constant) t = time
Uniformly Accelerated Motion – the acceleration of the particle is constant.
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Equations for Uniformly Accelerated Motion (Note: v = final velocity; v0 = initial velocity; a = acceleration; t = time; x = final position; x0 = initial position) - relates v and t - relates x and t - relates v and x FREELY FALLING BODY – one of the applications of uniformly accelerated motion Assumptions: All bodies in free fall near the earth’s surface have the same downward acceleration – acceleration due to gravity (g = 9.8 m/s2 = 32 ft/s2) 2. Air resistance is neglected. Note: Replace a with –g (Equations for Uniformly Accelerated Motion) on solving freely falling body. Sample Problem (Uniform Motion and Uniformly Accelerated Motion): A stone is dropped from a window 64 ft above the ground. How long does it take the ball to reach the ground? (b) What is its final velocity? Solution: Assume that the ground is the reference point. Assume also that the acceleration of the stone is 32ft/s2 (neglecting the air resistance). Given: y0 = 64 ft a. To compute for time, use the formula
where y=0 (ground) and v0=0
(initial velocity)
b. To compute for the final velocity, use the formula
or
PROJECTILE MOTION - refers to the motion of an object that is projected into the air horizontally or at an angle (assumption: motion occurs near the earth’s surface) - combination of horizontal and vertical motion (2D) analyzed separa(independent of each other). Equations Involved in Projectile Motion (Note: v0 = initial velocity; ₱ = angle measured above the horizontal)
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x-component (constant velocity)
y-component (constant acceleration)
Acceleration Initial Velocity
Final Velocity
Position v Points to Remember on Projectile Motion Air resistance is neglected. At maximum height, vertical velocity is zero while the horizontal velocity is equal to the initial velocity at the x-component. When a projectile returns to the height at which it was launched, its speed is equal to its initial speed but the direction of the y-component is now downward whereas the x-component remains the same. The total time of flight of the projectile (full trajectory/path of the projectile) is
Note that half of the total time is the time it takes a projectile to reach its maximum height. Maximum range of a projectile can be computed as
Note: θ = 450 has the maximum range. If ₱1 is an angle other than 450 that corresponds to a range R, then another angle θ2 for the same range is given by
Sample Problem (Projectile): A toy rifle is fired at an angle of 300 above the horizontal. (a) If the pellet’s initial velocity is 40 ft/s, how far does it go? (b) What is its time of flight? Solution: a.
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b.
FREE BODY DIAGRAM– is a diagram showing the chosen body by itself, “free” of its surroundings, with vectors drawn to show the magnitudes and directions of all the forces applied to the body by the various other bodies that interact with it. Techniques in how to use FBD on problem solving involving Newton’s laws: Newton’s first and second laws apply to a specific body. Only forces acting on the body matter. Free-body diagrams are essential to help identify the relevant forces. Note: When a problem involves more than one body, you have to take the problem apart and draw a separate free-body diagram for each body. Example: Problem: A worker applies a constant horizontal force with magnitude 20 N to a box with mass 40 kg resting on a level floor with negligible friction. What is the acceleration of the box? Free body Diagram:
NEWTON’S LAWS OF MOTION 1st Law: “A body at rest will remain at rest and a body in motion will remain in motion at constant velocity in a straight line if no net force acts on it.” inertia – the tendency of an object to resist a change in its state of motion mass – quantitative measure of an object’s inertia 2nd Law: “The net force acting on a body is proportional to the mass of the body and to its acceleration; the direction of the force is the same as that of the body’s acceleration. Force = (mass)(acceleration)
Units of Mass, Acceleration, and Force Units
Mass
Acceleration
Force 2
2
SI (mks)
kilogram (kg)
meter/(second) (m/s )
Newton (N)
Cgs
gram (g)
centimeter/(second)2
dyne
English
Slugs
Feet/(second)2(ft/s)2
Pounds
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Weight and Mass The weight of a body is the gravitational force with which the earth attracts the body. The weight of a body varies with its location near the earth (or other astronomical body), whereas its mass is the same everywhere in the universe. Weight: W = mg 3rd Law: “When one body exerts a force on another body, the second exerts an equal force in the opposite direction on the first.” Thus for every action force, there is an equal and opposite reaction force; no force can occur all by itself. Action and reaction forces never balance out because they act on different bodies. Sample Problem (Newton’s Law of Motion): a. What is the weight of an object whose mass is 5 kg? 0 b. What is its acceleration when a net force of 100 N acts on it? Solution: a. b.
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