Vectors 1D kinematics 2D kinematics Newton’s laws of motion Review for Test 1 Vectors 1D kinematics 2D kinematics Newton’s laws of motion

Vectors Any quantity that can be fully described by magnitude and direction. Examples: force, acceleration, momentum, displacement, velocity Can be added graphically (polygon method) and analytically (component method).

Polygon method Using the scale (e.g. 1 cm : 10 N) that is provided, draw the first vector using an arrow. Draw the second vector from the tip of the first vector. Using a third arrow, connect the tail of the first vector to the arrow head of the second vector. This is the resultant. The arrow head of the resultant is where the arrow head of the second vector is located. Using the scale, measure the length of the resultant to determine its magnitude. Use a protractor to determine the direction of the resultant. The direction of the resultant is the angle that its tail make with the x- of y-axis.

Vector Addition: Polygon Method B R A

Vector Addition: Component Method Say, you are to add vectors A and B. Sketch vector A. Calculate the x- and y-components of vector A. Don’t forget to include the correct sign. [sign is positive for x-component that is pointing to the right and negative for x-component that is pointing to the left. sign is positive for the y-component that is pointing upward and negative for y-component that is pointing downward]

Vector Addition: Component Method 3. Calculate the summation of x-components and the summation of y-components. The summation of the x-components is the x-component of the resultant R; the summation of the y-components is the y-component of the resultant R. 𝑥=𝑅𝑥=𝐴𝑥+𝐵𝑥 𝑦=𝑅𝑦=𝐴𝑦+𝐵𝑦

Vector Addition: Component Method 4. Based on the signs of Rx and Ry, sketch Rx using an arrow and then, using another arrow, sketch Ry from the arrow head of Rx. 5. Using a third arrow, connect the tail of Rx to the arrow head of Ry. This third arrow is the resultant R. 6. Since Rx, Ry and R make a right triangle, use the Pythagorean theorem to determine the magnitude of R. 𝑅= 𝑅𝑥 2 + 𝑅𝑦 2

Vector Addition: Component Method 7. Use the tangent function to determine the angle that the tail of R makes with the x- or the y-axis. This angle will be the orientation of R. 𝜃= 𝑡𝑎𝑛 −1 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 8. To complete the direction of R, specify how the angle was measured [e.g. N of E, S of E, W of N, above –x-axis, etc.] 9. Combine the result of step 6 and 8 to give a complete description of R [e.g. 750 m, 35 S of E].

Motion Essential Knowledge 3.A.1: An observer in a particular reference frame can describe the motion of an object using such quantities as position, displacement, distance, velocity, speed, and acceleration. a. Displacement, velocity, and acceleration are all vector quantities. b. Displacement is change in position. Velocity is the rate of change of position with time. Acceleration is the rate of change of velocity with time. Changes in each property are expressed by subtracting initial values from final values. c. A choice of reference frame determines the direction and the magnitude of each of these quantities.

Motion graph Sample motion graph for objects in uniform motion v t a t

Motion graph Sample motion graphs of objects in uniformly accelerated motion. v t a t

Motion graph The slope of the position vs time graph is equal to the velocity. The slope of the velocity vs time graph is equal to the acceleration. The area under the acceleration vs time graph is equal to the change in velocity. The area under the velocity vs time graph is equal to the displacement.

Kinematics equations 𝑣 𝑥 = 𝑣 0𝑥 + 𝑎 𝑥 𝑡 𝑥− 𝑥 0 = 𝑣+ 𝑣 0𝑥 2 𝑡 𝑥− 𝑥 0 = 𝑣 0𝑥 𝑡+ 1 2 𝑎 𝑥 𝑡 2 𝑣 𝑥 2 = 𝑣 0𝑥 2 +2 𝑎 𝑥 (𝑥− 𝑥 0 )

Free Fall Motion is under the influence of gravity only. Uniformly accelerated. Acceleration is equal to –g (-9.8 m/s2) Acceleration is always downward.

Projectile Motion A projectile is any object that once projected or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity. Horizontal motion is uniform motion. Vertical motion is uniformly accelerated motion (acceleration is –g = -9.8m/s2)

Projectile Motion The kinematics equations apply. Special case equations: 𝑡 𝑓𝑙𝑖𝑔ℎ𝑡 = 2 𝑣 0 𝑠𝑖𝑛 𝑔 𝑦− 𝑦 0 𝑚𝑎𝑥 = 𝑣 0 2 𝑠𝑖𝑛 2  2𝑔 𝑥− 𝑥 0 𝑚𝑎𝑥 = 𝑣 0 2 𝑠𝑖𝑛2 𝑔

Forces Essential Knowledge 3.A.2: Forces are described by vectors. a. Forces are detected by their influence on the motion of an object. b. Forces have magnitude and direction.

Forces Essential Knowledge 3.B.1: If an object of interest interacts with several other objects, the net force is the vector sum of the individual forces.

Force Essential Knowledge 3.A.3: A force exerted on an object is always due to the interaction of that object with another object. a. An object cannot exert a force on itself. b. Even though an object is at rest, there may be forces exerted on that object by other objects. c. The acceleration of an object, but not necessarily its velocity, is always in the direction of the net force exerted on the object by other objects.

Force Essential Knowledge 3.A.4: If one object exerts a force on a second object, the second object always exerts a force of equal magnitude on the first object in the opposite direction.

Weight The amount of gravitational force on a mass 𝐹 𝑔 =𝑚𝑔

Mass Essential Knowledge 1.C.1: Inertial mass is the property of an object or a system that determines how its motion changes when it interacts with other objects or systems. Essential Knowledge 1.C.2: Gravitational mass is the property of an object or a system that determines the strength of the gravitational interaction with other objects, systems, or gravitational fields. a. The gravitational mass of an object determines the amount of force exerted on the object by a gravitational field. b. Near the Earth’s surface, all objects fall (in a vacuum) with the same acceleration, regardless of their inertial mass. Essential Knowledge 1.C.3: Objects and systems have properties of inertial mass and gravitational mass that are experimentally verified to be the same and that satisfy conservation principles.

Inertial Mass vs Gravitational Mass Gravitational mass is defined by the force of gravitation, which states that there is a gravitational force between any pair of objects, which is given by  F = G m1 m2/r2  where G is the universal gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.  Inertial mass is defined by Newton's second law , F = ma, which states that when a force F is applied to an object, it will accelerate proportionally, and that constant of proportion is the mass of that object. 

Inertial Mass vs Gravitational Mass Physically, no difference has been found between gravitational and inertial mass. The only difference that we can find between inertial and gravitational mass that we can find is the method.  Inertial mass is found by applying a known force to an unknown mass, measuring the acceleration, and applying Newton's Second Law, m = F/a. Gravitational mass is measured by comparing the force of gravity of an unknown mass to the force of gravity of a known mass.

Free-body diagram Essential Knowledge 3.B.2: Free-body diagrams are useful tools for visualizing forces being exerted on a single object and writing the equations that represent a physical situation. a. An object can be drawn as if it was extracted from its environment and the interactions with the environment identified. b. A force exerted on an object can be represented as an arrow whose length represents the magnitude of the force and whose direction shows the direction of the force. c. A coordinate system with one axis parallel to the direction of the acceleration simplifies the translation from the free-body diagram to the algebraic representation.

Newton’s Law of Motion First law: Law of Inertia An object at rest will remain at rest, while an object in motion will continue moving at constant velocity, unless acted upon by an unbalanced external force.  𝐹 𝑥 =0; 𝐹 𝑦 =0 Inertia is the tendency of an object to resist change in its motion. Mass is the measure of inertia. The more mass an object has, the higher is its inertia.

Newton’s Laws of Motion Second Law: Law of Acceleration The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.  𝐹 𝑥 =𝑚 𝑎 𝑥 ;  𝐹 𝑦 =𝑚 𝑎 𝑦 The direction of the acceleration is the same as the direction of the net force.

Newton’s Laws of Motion Third Law: Law of Acceleration For every action, there is an equal but opposite reaction. Action-reaction forces act on different bodies. Example: If the action is the force of the earth on a book, then the reaction will be the force of the book on the earth. Action-reaction forces are always equal in magnitude but opposite in direction no matter what the situation. Forces always come in pairs; no force is isolated.

Gravitation Essential Knowledge 2.B.1: A gravitational field 𝑔 at the location of an object with mass m causes a gravitational force of magnitude mg to be exerted on the object in the direction of the field. a. On the Earth, this gravitational force is called weight. b. The gravitational field at a point in space is measured by dividing the gravitational force exerted by the field on a test object at that point by the mass of the test object and has the same direction as the force. c. If the gravitational force is the only force exerted on the object, the observed free-fall acceleration of the object (in meters per second squared) is numerically equal to the magnitude of the gravitational field (in newtons/kilogram) at that location.

Gravitation Essential Knowledge 2.B.2: The gravitational field caused by a spherically symmetric object with mass is radial and, outside the object, varies as the inverse square of the radial distance from the center of that object. a. The gravitational field caused by a spherically symmetric object is a vector whose magnitude outside the object is equal to 𝐺 𝑀 𝑟 2 . b. Only spherically symmetric objects will be considered as sources of the gravitational field.

Gravitation Gravitational force can be calculated using the formula: 𝐹 𝑔 =𝐺 𝑚 1 𝑚 2 𝑟 2 G = 6.67 x 10-11 m3/kgs2

Uniform Circular Motion An object moves at constant speed around a circular path. The centripetal acceleration is directed towards the center of the circular path. The centripetal acceleration is caused by a centripetal force. The angular velocity is constant.

Circular motion = 𝑠 𝑟 =  𝑡 𝑣= 𝑠 𝑡 𝑎 𝑐 = 𝑣 2 𝑟 =𝑟  2 𝐹 𝑐 =𝑚 𝑎 𝑐 Remember: 360 = 1 rev = 2rad

How to set up equations involving centripetal force Draw the free-body diagram (Draw only the actual force) Set up Newton’s second law of motion equation. If the object moves in a horizontal circle, then Fx = mac. If the object moves in a vertical circle, the Fy = mac. The acceleration ac is either expressed as 𝑣 2 𝑟 or 𝑟  2 .