AP physics final
2.2 Vectors, Scalars, and Coordinate Systems
-A vector is any quantity that has magnitude and direction -A scalar is any quantity that has magnitude but no direction. -displacement and velocity are vectors, whereas distance and speed are scalars -in one-dimensional motions, direction is specified by a plus or minus sign to signify left or right, up or down, and the like.
4.3 Newton's Second Law of Motion: Concept of a System
-Acceleration is defined as a change in velocity, meaning a change in its magnitude or direction, or both. -An external force is one acting on as system from outside the system, as opposed to internal forces, which act between components within the system. -Newton's second law of motion states that the acceleration of a system is directly proportional to and int he same direction as the net external force acting on the system, and inversely proportional to its mass. -in equation form: a = F(net) / m or F(net) = ma -the weight of an object is defined as the force of gravity acting on an object of mass. The object experiences an acceleration due to gravity: w = mg -if the only force acting on an object is due to gravity, the object is in free fall. -friction is a force that opposes the motion past each other of objects that are touching.
1.3 Accuracy, Precision, and Significant Figures
-Accuracy of a measured value refers to how close a measurement is to the correct value. The uncertainty in a measurement is an estimate of the amount by which the measurement result may differ from this value. -Precision of measured values refers to how close the agreement is between repeated measurements. -The precision of a measuring tool is related to the size of its measurement increments. The smaller the measurement increment, the more precise the tool. -Significant figures express the precision of a measuring tool. -when multiplying or dividing measured values, the final answer can contain only as many significant figures as the least precise value. -when adding or subtracting measured values, the final answer cannot contain more decimal places than the least precise value.
2.1 Displacement
-Kinematics is the study of motion without considering its causes. It is limited to motion along a straight line, called one dimensional motion. - Displacement is the change in position of an object. -in symbols, displacement Δx is defined to be Δx = x(final) - x(initial) where x(initial) is the initial position and x(final) is the final position. Δ (delta) always means "change in" whatever quantity follows it. The SI unit for displacement is the meter (m). Displacement has a direction as well as a magnitude. -When you start a problem, assign which direction will be positive. -Distance is the magnitude of displacement between two positions. -Distance traveled is the total length of the path traveled between two positions.
4.2 Newton's First Law of Motion: Inertia
-Newton's first law of motion states that a body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net external force. this is also known as the law of inertia. -inertia is the tendency of an object to remain at rest or remain in motion. Inertia is related to an object's mass. - Mass, is the quantity of matter in a substance.
4.7 Further Applications of Newton's Laws of Motion
-Newton's laws of motion can be applied in numerous situations to solve problems of motion. -some problems will contain multiple force vectors acting in different directions on an object. Be sure to draw diagrams, resolve all force vectors into horizontal and vertical components, and draw a free-body diagram. Always analyze the direction in which an object accelerates so that you can determine whether net force = ma or net force = 0. -the normal force on an object is not always equal in magnitude to the weight of the object. If an object is accelerating, the normal force will be less than or greater than the weight of the object. Also, if the object is on an inclined plane, the normal force will always be less that the full weight of the object. -some problems will contain various physical quantities, such as forces, acceleration, velocity, or position. You can apple concepts from kinematics and dynamics in order to solve these problems of motion.
4.4 Newton's Third Law of Motion: Symmetry in Forces
-Newton's third law of motion represents a basic symmetry in nature. It states: whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that the first body exerts. -a thrust is a reaction force that pushes a body forward in response to a backward force. Rockets, airplanes, and cars are pushed forward by a thrust reaction force.
6.5
-Newton's universal law of gravitation: every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In equation form it is: F = G(mM/r^2) where F is the magnitude of the gravitational force. G is the gravitational constant, given by G = 6.673 x 10^-11 m^2/kg^2 -Newton's law of gravitation applies universally.
1.2 Physical Quantities and Units
-Physical quantities are a characteristic or property of an object that can be measured or calculated from other measurements. -Units are standards for expressing and comparing the measurement of physical quantities. All units can be expressed as combinations of four fundamental units. -The four fundamental units are the meter (for length), the kilogram (for mass), the second (for time), and the ampere (for electric current). These units are part of the metric system, which uses powers of 10 to relate quantities over the vast ranges encountered in nature -the four fundamental units are abbreviated as follows: meter, m; kilogram, kg; second, s; and ampere, A. The metric system also uses a standard set of prefixes to denote each order of magnitude greater than or lesser than the fundamental unit itself. -unit conversions involve changing a value expressed in one type of unit to another type of unit. This is done by using conversion factors, which are ratios relating equal quantities of different units.
1.1 Physics: An Introduction
-Science seeks to discover and describe the underlying order and simplicity in nature -Physics is the most basic of the sciences, concerning itself with energy, matter, space and time, and their interactions -Scientific laws and theories express the general truths of nature and the body of knowledge they encompass. These laws of nature are rules that all natural processes appear to follow.
2.6 Problem-Solving Basics for One-Dimensional Kinematics
-The six basic problem solving steps for physics are: 1. examine the situation to determine which physical principles are involved. 2. make a list of what is given or can be inferred from the problem as stated (identify the known). 3. identify exactly what needs to be determined in the problem (identify the unknown). 4. find an equation or set of equations that can help you solve the problem. 5. substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units. 6. check the answer to see if it is reasonable: Does it make sense?
2.5 Motion Equations for Constant Acceleration in One Dimension
-To simplify calculations we take acceleration to be constant, so that α(average) = α -We also take initial time to be zero -initial position and velocity are given a subscript 0; final values have no subscript. thus, Δt = t Δx = x - x(initial) Δv = v - v(initial) The following kinematic equations for motion with constant a are useful; x = x(initial) + vt v(average) = [v(initial) + v] / 2 v = v(inital) = at x= x(inital) + v(intial)t + 1/2at^2 v^2 = v(intial)^2 + 2a[x - x(initial)] -in vertical motion, y is substitued for x
2.4 Acceleration
-acceleration is the rate at which velocity changes. In symbols, average acceleration is α(average)=Δv/Δt = [v(final) - v(initial)] / [t(final) - t(initial)] -The SI unit for acceleration is m/s^2 -Acceleration is a vector, and thus has a both a magnitude and direction. -Acceleration can be caused by either a change in the magnitude or the direction of the velocity -instantaneous accelerations is the acceleration at a specific instant in time -deceleration is an acceleration with a direction opposite to that of the velocity
2.7 Falling Objects
-an object in free-fall experiences constant acceleration if air resistance is negligible. -on earth, all free-falling objects have an acceleration due to gravity, which averages g = 9.80 m/s^2 - whether the acceleration should be taken as +g or -g is determined by your choice of coordinate system. If you choose the upward direction as positive, a = -g = -9.8 m/s^2 is negative. In the oppisite case, a = +g = 9.8 m/s^2 is positive. Since acceleration is constant, the kinematic equations above can be applied with the appropriate +g or -g substituted for a. -For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration.
6.2 Centripetal Acceleration
-centripetal acceleration is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity and has the magnitude a(centripetal) = v^2/r or a(centripetal) = rω^2 -the unit or centripetal acceleration is m/s^2
6.3 Centripetal Force
-centripetal force is any force causing uniform circular motion. It is a "center-seeking" force that always points toward the center of rotation. It is perpendicular to linear velocity and has magnitude F(centripetal) = ma(centripetal) which can also be expressed as F(centripetal) = mv^2/r or F(centripetal) = mrω^2
4.1 Development of Force Concept
-dynamics is the study of how forces affect the motion of objects. -force is a push or pull that can be defined in terms of various standards, and it is a vector having both magnitude and direction. -external forces are any outside forces that act on a body. a free-body diagram is drawing of all external forces acting on a body
5.1 Friction
-friction is a contact force between systems that opposes the motion or attempted motion between them. Simple friction is proportional to the normal force pushing the systems together. Friction depends on both of the materials involved. The magnitude of static friction between systems stationary relative to one another is given by f(static) is less than or equal to μ(static)N where μ(s) is the coefficient of static friction, which depends on both of the materials. -the kinetic friction force between systems moving relative to one another is given by f(kinetic) = μ(kinetic)N where μ(k) is the coefficient of kinetic friction, which also depends on both materials.
2.8 Graphical Analysis of One-Dimensional Motion
-graphs of motion can be used to analyze motion. -graphical solutions yield identical solutions to mathematical methods for deriving motion equations. -the slope of a graph of displacement 'x vs. time' is velocity. -the slope of a graph of velocity vs. time is acceleration. -average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.
3.4 Projectile Motion
-projectile motion is the motion of an object through the air that is subject only to the acceleration of gravity. -to solve projectile motion problems, perform the following steps: 1.Determine a coordinate system. Then, resolve the position and/or velocity of the object in the horizontal and vertical components. The components of position are given by the quantities of x and y, and the components of the velocity are given by v(subscript x) = vcosθ and v(subscript y) = vsinθ, where v is the magnitude of the velocity and θ is its direction. 2. Analyze the motion of the projectile in the horizontal direction using the following equations: Horizontal motions [a(subscript x) = 0] x = x(initial) + v(sub x)t v(sub x) = v(initial sub x) = v(sub x) = velocity is a constant 3. Analyze the motion of the projectile in the vertical direction using the following equations: Vertical motion (Assuming positive direction is up; a(sub y) = -g = -9.80 m/s^2) y = y(initial) + 1/2[v(initial sub y) + v(sub y)]t v(sub y) = v(initial sub y) - gt y = y(initial) + v(initial sub y)t - 1/2gt^2 v(sub y)^2 = v(initial sub y)^2 - 2g[y - y(initial)] 4. recombine the horizontal and vertical components of location and/or velocity using the following equations: s = (square root) x^2 + y^2 θ = tan ^-1 (y/x) v = (square root) v(sub x)^2 + v(sub y)^2 θ(sub v) = tan^-1 [v(sub y)/v(sub x)] -The maximum height of a projectile launched with initial vertical velocity is given by h = v(inital vertical)^2/ 2g -The maximum horizontal distance traveled by a projectile is called the range. the range of a projectile on level ground launched at an angel θ(initial) above the horizontal with initial speed is given by R = [v(inital)^2 sin2θ(initial)] / g
3.1 Kinematics in Two Dimensions: An Introduction
-the shortest path between any two points is a straight line. In two dimensions, this path can be represented by a vector with horizontal and vertical components. -the horizontal and vertical components of a vector are independent of one another. Motion in the horizontal direction does not affect motion in the vertical direction, and vice versa.
2.3 Time, Velocity and Speed
-time is measured in terms of change, and its SI unit is the second (s). Elapsed time for an event is Δt = t(final) - t(initial) where t(final) is the final time and t(initial) is the initial time. The initial time is often taken to be zero, as if measured with a stopwatch; the elapsed time is then just t. -average velocity is defined as displacement divided by the travel time. In symbols, average velocity is v(w/line above) = Δx/Δt = [x(final) - x(initial)] / [t(final) - t(intial)] -the SI unit for velocity is m/s -Velocity is a vector and thus has a direction -Instantaneous velocity, v, is the velocity at a specific instant or the average velocity for an infinitesimal interval. -instantaneous speed is the magnitude of the instantaneous velocity -instantaneous speed is a scalar quantity, as it has no direction specified -average speed is the total distance traveled divided by the elapsed time. (Average speed is not the magnitude of the average velocity). Speed is a scalar quantity; it has no direction associated with it.
4.6 Problem Solving Strategies
-to solve problems involving Newton's laws of motion, follow the procedure described: 1. draw a sketch of the problem 2. identify known and unknown quantities, and identify the system of interest. Draw a free-body diagram. If vectors act in directions that are not horizontal or vertical, resolve the vectors into horizontal and vertical components and draw them on the free-body diagram 3. write Newton's second law in the horizontal and vertical directions and add the forces acting on the object. If the object does not accelerate in a particular direction the net force (horizontal) = 0. if the object does accelerate in that direction, net force (horizontal) = ma 4. check your answer. Are the units correct?
6.1 Rotation Angle and Angular Velocity
-uniform circular motion is motion in a circle at constant speed. The rotation angle Δθ is defined as the ratio of the arc length to the radius of curvature: Δθ = Δs/r where arc length Δs is the distance traveled along a circular path and r is the radius of curvature of the circular path. The quantity Δθ is measured in units of radians (rad), for which 2πrad = 360 degrees = 1 revolution - the conversion between radians and degrees is 1 rad = 57.3 degrees -angular velocity is the rate of change of an angle, ω = Δθ/Δt where a rotation Δθ takes place in a time Δt. the units of angular velocity are radians per second (rad/s). Linear velocity and angular velocity are related by v = rω or ω = v/r
4.5 Normal, Tension, and Other Examples of Forces
-when objects rest on a surface, the surface applies a force to the object that supports the weight of the object. This supporting force acts perpendicular to and away from the surface. It is called normal force. -when objects rest on a non-accelerating horizontal surface, the magnitude of the normal force is equal to the weight of the object: N = mg -when objects rest on an inclined plane that makes an angle with the horizontal surface, the weight of the object can be resolved into components that act perpendicular and parallel to the surface of the plane. These components can be calculated using: w(parallel) = w sin(θ) = mg sin(θ) w(perpendicular) = w cos(θ) = mg cos(θ). -the pulling force that acts along a stretched flexible connector, such as a rope or cable, is called tension. When a rope supports the weight of an object that is at rest, the tension in the rope is equal to the weight of the object: T = mg -in any inertial frame of reference (one that is not accelerated or rotated), Newton's laws have the simple forms given and all forces are real forces having a physical origin
1.4 Approximation
Scientists often approximate the values of quantities to perform calculations and analyze systems.