1.1: Units - Various systems of units developed - British (imperial) system - Length - foot (ft) - Weight - pound (lbs) - Time - second (s) - Mass - slug - Metric system (most common) - Meters, kilograms, and seconds (MKS) or centimeters, grams, and seconds (CGS) - SI units include MKS system + others in the table below - - Each system has base units and derived units - Base units: standard units that the system is built around - Derived units: created by associating base units with each other (i.e. N = kg*m/s^2) Prefixes can be used in the metric system can convert between imperial and metric system (See chapter 10) At the molecular, atomic, or subatomic level, different units may be used to work with the very small size - Angstroms: 1Å = 10-10 m - nanometers : 1 nm = 10-9 m - Electron volts: 1 eV = 1.6*10-19 J - Represents the amount of kinetic energy gained by an electron accelerating through a potential difference 1.2: Vectors and Scalars - Vectors: numbers that have magnitude and direction - i.e. displacement, velocity, acceleration, and force - Scalars: numbers that have magnitude only and NO direction - - - - i.e. Distance, speed, energy, pressure, and mass Vectors can be represented by arrows (represented by arrowhead over a letter or bold letter) - The direction of the arrow indicates the direction of the vector - Length of arrow ~ proportional to the magnitude of the vector quantity The magnitude of the vector represented by absolute value signs around the letter with the arrowhead or around the bold letter or italic type Scalars and magnitude of vectors are usually represented with italic type Vector addition - Resultant: the sum or difference of two or more vectors - Tail to tip method - Place the tail of B on the tip of A without changing the length or direction of the arrow - Lengths of the arrow must be proportional to the magnitude - Vector sum = vector joining the tail of A to the tip of B and pointing toward the tip of B Component method - Break each vector into perpendicular components - Usually horizontal and vertical (x- and y-components) but can be defined as the perpendicular and parallel components - Given any vector, find the x and y component by drawing a right triangle with the vector as the hypotenuse - - If theta is the angle between the vector and the x-component then - Cos theta = X/V → X = VCostheta - Sin theta = Y/V → Y = VSintheta If X and Y are already known, V can be found using the Pythagorean theorem: X2 + Y 2 = V2 or V = - - 2 𝑋 +𝑌 2 Angle of the resultant vector can be calculated using theta = arctan (Y/X) - NOTE: DO NOT NEED TO KNOW HOW TO CALCULATE ARCTAN IN MY HEAD - X component of the resultant vector is the sum of all the x components and the y component of the resultant vector is the sum of all the y components - Steps to component method - Resolve the vectors into X and Y components - Add the x components to get the X component of the resultant vector - Add the y components to get the y component of the resultant vector - Find the magnitude of the resultant vector using the Pythagorean theorem - Find the direction (theta) of resultant vector using theta = arctan (Y/X) Vector subtraction - - - Subtracting one vector from another can be done by adding a vector with equal magnitude but opposite direction to the first vector - Can be performed on the component vectors first then combined to create the resultant vector - X component of resultant vector is the difference between the x components of the vectors being subtracted - Y component of resultant vector is the difference between the y components of the vectors being subtracted Multiplying vectors by scalars - When a vector is multiplied by a scalar: - the magnitude of the vector will change ( new vector = old vector*factor) - Direction will be parallel or antiparallel to its original direction - Positive value → same direction - Negative value → opposite direction Multiplying a vector by other vectors - Use two vector quantities to generate a third vector or a scalar - To get a scalar (commutative) - Multiply the magnitudes of two vectors and the cosine of the angle between the two vectors → dot product - A •B = |A| |B| Cos theta - To get a third vector (noncommutative) - Multiply magnitude of two vectors and the sine of the angle between the two vectors → cross product - A ×B = |A| |B| Sin theta - Resultant vector will be perpendicular to the plane created by the two vectors → vector will be going into or out of the page - Finding the direction of resultant vector from the cross product - Right-hand rule - Point thumb in the direction of vector A - Extend fingers into the direction of vector B - Palm points in direction of the new vector 1.3: Displacement and Velocity - Displacement: The change in position that an object experiences - Vector quantity → magnitude and direction - Connects (in a straight line) the object's initial and final position - Does NOT account for the actual pathway taken by the object! ONLY accounts for net change (final - initial) - Distance: scalar quantity that considers the pathway taken by the object - - Velocity: vector quantity. Magnitude measures the rate of change of displacement in a given unit of time - Units: m/s - Direction of the velocity vector is in the direction of displacement Speed: rate of actual distance traveled in a given unit of time (units: m/s) Instantaneous speed of an object will always equal the magnitude of the object’s instantaneous velocity Instantaneous velocity: measure of the average velocity as the change in time approaches zero - Instantaneous speed is a scalar quantity Average speed will not always be equal to the magnitude of average velocity bc average speed is total distance over time while average velocity is displacement over time - Average velocity can be found as: 1.4: Forces and Acceleration - Every change in velocity is caused by a push or pull (a force) - Force: vector quantity that is experienced as pushing or pulling on objects (Units: N = 1 kg*m/s2) - Can exist between objects that are not touching - Gravity: attractive force felt by all forms of matter - All object exert gravitational forces on each other - Gravitational forces are usually insignificant on the small scale bc other forces have larger effect but are really important on the planetary scale - Magnitude of gravitational force between two objects (two masses) (universal law of gravitation) - - G = universal gravitational constant (6.67 * 10 -11 N*m2/kg2) - m1 and m2 = masses - r = distance between the centers of mass Friction - Friction: force that opposes the movement of an object - Friction always opposes an object's motion and causes it to slow down or become stationary - Static friction(fs): exists between a stationary object and the surface it is on - μs = coefficient of static friction - Unitless quantity that is dependent on the two materials in contact N = magnitude of the normal force - Force that is perpendicular to the surface that the object rests on - - - - Less than or equal to sign indicates that there is a range of values for static friction - Minimum is zero (on surface with no applied forces) - Max is μs*N Kinetic Friction (fk): exists between a sliding object and the surface over which the object slides - Only present when the object is sliding! Be careful to identify if the object is sliding or not - Magnitude of kinetic friction: - μK = coefficient of kinetic friction - N = normal force - Kinetic friction has a constant value given by the equation Coefficient of static friction is always larger than coefficient of kinetic friction, so the maximum value for static friction is always greater than the value of kinetic friction - Harder to get an object sliding than it is to keepit sliding - Once you get the object sliding it is easier to maintain the motion Mass and weight - Mass(m): measure of the body’s inertia or amount of matter in the object. Constant value. - Scalar quantity so only magnitude - Units: Kg - Weigh(Fg)t: measure of gravitational force on an object’s mass - Vector quantity → magnitude and direction - Units: N - Weight and mass are related: Fg = mg - Fg = weight of object - m = mass of object - g = acceleration due to gravity (9.8 m/s2 for earth) - Weight of object is applied at center of mass/gravity - Each point of an extended object moves through space in its own path, but one point in the extended object moves in a simple parabolic path (center of mass) - Center of mass is related to a single point where one can visualize gravity acting on an object - Center of mass for a uniform object is at the geometric center, but the same is not true for nonsymmetric, nonuniform objects - - Calculate the coordinate x,y and z if needed using the mass and distance that mass is from the specified origin - Acceleration (a): rate of change of velocity that an object experiences as a result of some applied force - Vector quantity → magnitude and direction - Units: m/s2 When acceleration is in the opposite direction of initial velocity → acceleration - Average acceleration: Instantaneous velocity: average acceleration as Δt approaches 0 → - On a velocity vs time graph, the slope of the graph at a specific time → instantaneous acceleration - Positive slope → positive acceleration and in same direction as velocity - Negative slope → negative acceleration (deacceleration) and in opposite direction of velocity 1.5: Newton’s Laws - First law (law of inertia) - Fnet = ma = 0 - Fnet = net force - m = mass - a = acceleration - A body either at rest or in motion with a constant velocity (no acceleration) will remain that way unless a net force acts upon it - Second law - Fnet = ma - Fnet = net force - m = mass - a = acceleration - An object of mass m will accelerate when the vector sum of the forces results in some nonzero resultant force vector - Net force and acceleration point in the same direction - The direction of net force determines the direction of acceleration - Third law (action and reaction) - FAB = -FBA - The force exerted by A on B is equal but opposite to the force exerted by B on A - Forces between two objects are equal but opposite 1.6: Motion With Constant Acceleration - Object can undergo two types of motion - Constant (no acceleration) - Changing (acceleration) - Acceleration can be constant or changing - Constant acceleration is important - Linear motion: objects velocity and acceleration are along the line of motion → pathway of moving object continues along a straight line - Linear motion problems usually involve an object being dropped to the ground from a height - Falling objects exhibit linear motion with constant acceleration which can be described by the following equations: - - - - x = displacement (change to y when the motion is vertical) - v = velocity - a = acceleration - V0 = initial velocity - t = time If no other force is acting on an object that is falling other than gravity, it is in free fall and will not reach terminal velocity Free fall questions - Use kinematics - If air resistance is negligible, the amount of time to reach max height = time to fall back down to starting height - Set vf = 0 and solve for t - Multiply t by 2 to find the total time in the air - Only force acting on object in air is gravity, so velocity in the x direction is constant - Multiply initial velocity in x direction by time in air to find the horizontal displacement Air resistance opposes the motion of the object - Value of air resistance grows as the speed of the object increases Object in free fall experiences a growing drag force (since air resistance increases with speed) Drag force will eventually equal the magnitude of the weight of the object → object falling with constant velocity (no net force → no acceleration) Terminal velocity: the constant velocity of an object when the weight force = drag force Projectile motion - Projectile motion: motion that follows a path along two dimensions - Velocity and acceleration in the two dimensions are independent of each other (they do not influence each other) → analyze them separately - Gravity is only experienced in the y direction - Vy will change at the rate of g (9.8 m/s2) while Vx will remain constant - No forces act along the x-axis so the velocity is constant in the x direction - - Inclined Planes - Motion in two dimensions - Divide forces into vector components that are parallel and perpendicular to the plane - Gravity is usually split into vector (double check to make sure this fits the picture) - Fgy = mgsintheta - Fgx = mgcostheta - Same kinematics equations can be applied Circular motion - Occurs when forces cause an object to move in a circular pathway - Completion of one cycle → displacement of object = 0 - Uniform circular motion → speed of object is constant - nonuniform circular motion → speed is not constant - In uniform circular motion - Instantaneous velocity vector is always tangent to the circular path - Object has a tendency to break out of circular motion and move in a linear path along the tangent - Centripetal force keeps the object from breaking from circular pathway - Centripetal force = any force that points radially inward and causes circular motion - Forces can be split into radial and tangential components, but the tangential force is zero in uniform circular motion since there is no change in the speed of the object - Centripetal force → centripetal acceleration - Acceleration is always in the direction of the net force - For circular motion, acceleration is inward toward the center of the circle (direction of centripetal force) - Acceleration generated by centripetal force keeps the object in the circular pathway - When centripetal force is no longer acting on the object → object will exit circular path and travel along the tangent - Centripetal acceleration = v2/r - FC= ma = mv2/r 1.7: Mechanical Equilibrium - Free body diagrams used to represent forces acting on an object - Dynamics: the study of forces and torques - - Translational equilibrium - Translational motion occurs when forces cause an object to move without any rotations - Translational equilibrium exists only when the vector sum of all the forces acting on an object is zero (first condition of equilibrium) - If there is no net force, there is no acceleration - Means the object is at rest OR there is constant nonzero velocity - Objects in translational equilibrium will have constant velocity (no acceleration, so it could be zero velocity or same nonzero velocity) and constant direction Rotational equilibrium - Rotational motion occurs when forces are applied against an object in a way that causes the objects to rotate around a fixed pivot point (the fulcrum) - Applying force some distance from the pivot → torque (vector quantity) - Force at the pivot → no torque - Distance between applied force and the pivot = lever arm - Torque generates rotational motion - Torque depends on - the magnitude of force - distance from pivot the force is applied angle - Angle between force and lever arm - Sin (90) = 1 so greatest torque generated when applied perpendicular to lever arm Sin (0) = 0 so no torque generated when applied parallel to lever arm - Rotational equilibrium exists only when the vector sum of all the torques acting on an object is zero (second condition of equilibrium) Clockwise torque = negative Counterclockwise torque = positive When in rotational equilibrium, the object can - Be stationary (not rotating at all) - Rotate with constant angular velocity
