Equilibrium Forces and Unbalanced Forces

A force is a push or a pull applied to an object. Topic Overview A force is a push or a pull applied to an object. A net Force (Fnet) is the sum of all the forces on an object (direction determines + or -) Fnet = 6N to the right

Isaac Newton has 3 laws that describe the motion of object 1st Law: Law of inertia An object at rest will stay at rest unless acted on by an outside force Inertia: The amount of mass an object has More inertia = more mass = Harder to move

When an object is in equilibrium, the net force equals zero Topic Overview When an object is in equilibrium, the net force equals zero Equilibrium  Fnet = 0 Objects in equilibrium can either be at rest or be moving with constant velocity Up = Down Forces Left = Right Forces

3rd Law: Equal and Opposite For every action, there is an equal and opposite reaction. “Things push back”

Topic Overview When an object has unbalanced forces acting on it, the object will accelerate in the direction of that excess force: Fnet = ma This is called “Newton's Second Law”

Example Problem – Balanced Forces

Example Problem – Balanced Forces Solution

Example Problem – Unbalanced Forces

Example Problem – Unbalanced Forces - Solution

Make sure you know if the object is in equilibrium or not Common Mistakes Make sure you know if the object is in equilibrium or not

Circular Motion

Topic Overview An object in circular motion has a changing velocity but constant speed. This is possible because the objects speed does not change (same m/s) but the direction of its motion does change

The circular force (Fc ) is always directed toward the center Topic Overview The velocity of the object is always “tangent” to the path of the object. The circular force (Fc ) is always directed toward the center The acceleration is always toward the center of the circle Velocity Force

r is the radius of the circle Equations r is the radius of the circle

Example Problem

Example Problem - Solution

Be sure to square the velocity Cross multiply when solving for “r” Common Mistakes Be sure to square the velocity Cross multiply when solving for “r”

Momentum/Impulse

Momentum: The product of the mass and velocity of an object Momentum Recap Momentum: The product of the mass and velocity of an object Equation: p = mv Units: p = kilograms meters per second (kgm/s) Momentum is a vector: When describing the momentum of an object, the direction matters.

Momentum Before = Momentum After Momentum Recap Collision: When 2 or more objects interact they can transfer momentum to each other. Conservation of Momentum: The sum of the total momentum BEFORE a collision, is the same as the sum of the total momentum AFTER a collision Momentum Before = Momentum After p1i +p2i + p3i = p1f+p2f + p3f

To Solve Collision Problems: Momentum Recap To Solve Collision Problems: Step 1: Find the total momentum of each object before they interact Step 2: Set it equal to the total momentum after they collide Remember momentum is a vector, so you have to consider if the momentum is (+) or(-) when finding the total!!! Initial = Final 0 = -1.2 (v) + (1.8)(2) Initial = Final (1)(6)+ 0 = (1 + 3.0) v

Fnett = p = mv Units = Ns Impulse An outside force will cause a change in the momentum of an object. This is called an impulse. IMPULSE: A change in momentum Fnett = p = mv Units = Ns

Impulse To find the impulse under a force vs. time graph, you would find the area under the line. I = Ft

Example Problem - 1

Example Problem 1- Solution

Kinematics

Kinematics is what we use to describe the motion of an object. Topic Overview Kinematics is what we use to describe the motion of an object. We use terms such as displacement, distance, velocity, speed, acceleration, and time to describe the movement of objects.

Distance (m): The total meters covered by an object (odometer) Topic Overview Distance (m): The total meters covered by an object (odometer) Displacement (m): The difference between the start and end points.

How fast the car is moving is a (+) and (–) indicate direction Topic Overview Velocity (m/s): How fast the car is moving is a (+) and (–) indicate direction Acceleration (m/s2) Acceleration is a change in velocity If an object is accelerating, it is either speeding up or slowing down

x = vt x = vavet Some objects have constant velocity Topic Overview Some objects have constant velocity Again this means that acceleration = 0 The equation for constant velocity is: x = vt If you are given an average velocity (vave), it is the same thing as constant velocity: x = vavet

a = (vf-vi) / t x = vit + ½ at2 vf2 = vi2 + 2ax Topic Overview Some objects have constant acceleration This means they are speeding up or slowing down The equations for constant acceleration: a = (vf-vi) / t x = vit + ½ at2 vf2 = vi2 + 2ax

If an object is “at rest” Velocity = 0 Topic Overview To solve problems with constant acceleration, make a table! Vi Initial velocity Vf Final velocity a Acceleration x Distance t time Reminder: If an object is “at rest” Velocity = 0

All falling objects accelerate at 9.8m/s2 due to gravity Topic Overview One example of a “constant acceleration problem” is a falling object (an object traveling through the air) All falling objects accelerate at 9.8m/s2 due to gravity They also start with Zero initial velocity Vi 0m/s Vf a 9.8m/s2 x t

In this case, choose the equation that does not have vf Example Problem A stone is dropped from a bridge approximately 45 meters above the surface of a river.  Approximately how many seconds does the stone take to reach the water's surface? Reminder: Choose the equation that does not have the “blocked off” variable. In this case, choose the equation that does not have vf Vi 0m/s Vf a 9.8m/s2 x 45 t ? a = (vf-vi) / t x = vit + ½ at2 vf2 = vi2 + 2ax

Example Problem - Solution

Graphs

1) Displacement vs. Time graphs Topic Overview The motion of an object can be represented by three types of graphs (x, v, a) 1) Displacement vs. Time graphs Tells you where the object is The slope (steepness) is the velocity In the graph above A is faster than B A X (m) Time (s) B

1) Types of Motion for x vs. t graphs Topic Overview 1) Types of Motion for x vs. t graphs X (m) Time (s) X (m) Time (s) X (m) Time (s) Not moving because the position does not change Constant velocity because the slope does not change (linear) Accelerating because it is a curve

2) Velocity vs Time graphs Topic Overview 2) Velocity vs Time graphs Tells you how fast the object is moving Slope of the line = Acceleration Area under curve = Displacement v (m/s) (s)

2) Velocity vs. Time graphs Topic Overview 2) Velocity vs. Time graphs v (m/s) (s) v (m/s) (s) v (m/s) (s) Constant speed because the value for velocity does not change Speeding up because the value of the velocity is moving away from zero Slowing down because the value of the velocity is moving toward zero

Tells you the acceleration of the object Area under curve = Velocity Topic Overview 3) Acceleration vs. time Tells you the acceleration of the object Area under curve = Velocity a (m/s2) (s)

3) Acceleration vs. Time graphs Topic Overview 3) Acceleration vs. Time graphs a (m/s2) a (m/s2) a (m/s2) (s) (s) (s) No acceleration. This means the object has a constant speed This object has positive acceleration This object has negative acceleration

Example Question-2

Example Question-2 Solution

Calculation Examples:

Example Question-1 Displacement: Difference between y value Average Speed: Slope between 2 time points Instantaneous speed: Slope at 1 time point (TANGENT)

Example Question-1 Solution

What is the average speed from Example Question-2 What is the average speed from 0-6 seconds?

Example Question-2 Solution What is the average speed from 0-6 seconds? Average is the slope between 2 points Slope = (4 – 0) = 0.66m/s (6 – 0)

What is the displacement from 1-4 seconds? Example Question-3 What is the displacement from 1-4 seconds?

Example Question-3 Solution Position increased (+) from 2m to 4m Answer: 4 – 2 = 2m

Projectile Motion

Topic Overview Objects that travel in both the horizontal and vertical direction are called “projectiles”. These problems involve cars rolling off cliffs, objects flying through the air, and other things like that.

An object traveling through the air will: Topic Overview An object traveling through the air will: ACCELERATE in the VERTICAL DIRECTION Because it is pulled down by gravity Have CONSTANT VELOCITY in the HORIZONTAL Because there is no gravity Because they are different, we do calculations in the horizontal (x) and vertical (y)_ SEPARATELY.

Zero Launch Angle Projectile Motion Acceleration -9.8m/s2 VelocityHorizontal Constant VelocityVertical Increasing The time for an object to fall is determined by drop height ONLY (horizontal velocity has no effect)

x = vt a = (vf-vi) / t x = vit + ½ at2 vf2 = vi2 + 2ax Projectile Motion The key to solving projectile motion problems is to solve the horizontal and vertical parts SEPARATELY. Horizontal Constant Velocity x = vt Vertical Accelerating a = (vf-vi) / t x = vit + ½ at2 vf2 = vi2 + 2ax Time is the only thing that is the same! TIME = TIME

Example 1 A bullet is shot at 200m/s from a rifle that is 2.5m above the ground. How far downrange will the bullet reach before hitting the ground? Step 2: Use the time to find out the distance in the other direction (in this case the horizontal direction) Step 1: Find the time. In this case we have to use the vertical height to find the time. (acceleration) vi vf a -9.8 x 2.5 t ?? x = vit + ½ at2 x = vt x = 200 (0.72) x = 150m t = 0.72s

Energy and Power

Kinetic Energy: VELOCITY Potential Energy: HEIGHT Energy is the ability to CHANGE an object. These types of energy are a result of a CHANGE in……. Work: FORCE Kinetic Energy: VELOCITY Potential Energy: HEIGHT Elastic Energy: SHAPE

Conservation of Energy In a system, the TOTAL MECHANICAL ENERGY never changes Energy can switch forms, but it cannot be created or destroyed. WORK POTENTIAL KINETIC ELASTIC

Conservation of Energy Mathematically speaking that looks like this TME Initial = TME Final KE + PE + EE + W = KE + PE + EE + W

Energy: Joules (J) Work: W = Fd Kinetic Energy: KE = ½mv2 Potential Energy: PE = mgh Elastic Energy: EE = ½kx2

Potential Energy  Kinetic Energy Examples A ball is dropped from a height of 12m, what is the velocity of the ball when it hits the ground? Potential Energy  Kinetic Energy Since all of the energy is transferred, we cans set them equal to each other PE = KE m(9.8)(12) = ½(m)v2  mass cancels 15.33 m/s = v

Potential Energy  Kinetic Energy Examples A ball is dropped from a height of 12m, what is the velocity of the ball when it hits the ground? Potential Energy  Kinetic Energy Since all of the energy is transferred, we can set them equal to each other PE = KE m(9.8)(12) = ½(m)v2  mass cancels 15.33 m/s = v

Examples A force of 50N pushes horizontally on a 5kg object for a distance of 2m. What is the final velocity of the object? Work  Kinetic Energy Since all of the energy is transferred, we can set them equal to each other Work = KE (50)(2) = ½(5)v2 6.32 m/s = v

TME Initial = TME Final Examples No matter type of energy transfer, the set up is the same. Even if there is more than one type of energy present TME Initial = TME Final KE + PE + EE + W = KE + PE + EE + W

P = E/t Power = Energy/ time On your equation sheet it lists “Energy” as Work for the top of the fraction. But you can put any type of energy on the top part of this equation.

Regular Physics ONLY: Electrostatics

Atoms have protons, neutron, and electrons. Electrostatics: Atoms have protons, neutron, and electrons. Only electrons can move! 1 e- = 1.6E-19 Coulomb (C)

Electrostatics: An object becomes charged when its electrons are shifted or transferred Extra electrons = (-) Fewer electrons = (+)

Electrostatics: How many more electrons than protons are there on an object with a -1.76E-18 C charge? What is the total charge of an object with a deficiency of 4.0 x108 electrons? Extra electrons = (-) Fewer electrons = (+)

Two charged objects will have an Electric Field between them. Electrostatics: Two charged objects will have an Electric Field between them. Field Lines (+) (-)“Start Positive” 

Two charged objects will feel an Electric Force Electrostatics: Two charged objects will feel an Electric Force Opposite charges attract (+/-) Same charges repel (+/+) (-/-)

Calculating the Electric Force Electrostatics: Calculating the Electric Force k = 9E9 q=charge r= distance between the charges

Calculating the Electric Force Electrostatics: Calculating the Electric Force k = 9E9 What is the magnitude of the electrostatic force between two electrons separated by a distance of 1.00 × 10–8 meter?

Regular Physics ONLY: Circuits

Circuits: A circuit provides a COMPLETE path for electrons to move. The flow of electrons is called the current (I). Electrons flow because a voltage (V) provides an energy difference In order to get energy out of a circuit, there has to be resistors (R).

Circuits:

Series Circuits Circuits: A circuit in which there is only one current path

I = I1 = I2 = I3 = I4 V = V1 + V2 + V3 RT= R1 + R2 + R3 Series Circuit Circuits: Series Circuit Current is the same in all resistors I = I1 = I2 = I3 = I4 Voltage is distributed among the resistors V = V1 + V2 + V3 Total Resistance is the sum of all resistors. RT= R1 + R2 + R3

Parallel Circuits Circuits: A circuit in which there are several current paths

Circuits: Series or Parallel?????

VT = V1 = V2 = V3 IT = I1 + I2 + I3 Parallel Circuit Circuits: Parallel Circuit Current is the added in all resistors IT = I1 + I2 + I3 Voltage is equal among the resistors VT = V1 = V2 = V3 Total Resistance is the reciprocal of all resistors.3 1/RT = 1/R1 + 1/R2 + 1/R3

Circuits: RIVP TABLES!!!

AP ONLY: Torque

Units for torque: Nm (Newton-meters) Torque Introduction torque = distance from axis of rotation x force x sin (angle between r and F) Units for torque: Nm (Newton-meters)

Torque Torque Introduction Torque is analogous to a force. Force Torque Linear Acceleration Angular Acceleration Torque

Torque Introduction The “distance” of the force also matters. Lever Arm (r): The distance between the pivot point and the force.

The “angle (θ)” of the force also matters. Torque Introduction The “angle (θ)” of the force also matters. 10N 10N

The “angle” of the force also matters. Torque Introduction The “angle” of the force also matters. Use the sin component of the force Measure angle from the LEVER ARM  FORCE 10N 10N

Torque – Direction and equilibrium Net Force = Zero  No acceleration Net Torque = Zero  No rotation Clockwise = (+) Counterclockwise = (-) Is the object moving?

Ex 1 6kg 10kg To weigh a fish a person hangs a tackle box of mass 6 kilograms and a cooler of mass 10 kilograms from the ends of a uniform rigid pole that is suspended by a rope attached to its center. The system balances when the fish hangs at a point 1/4 of the rod’s length from the tackle box. What is the mass of the fish?

Ex 2 A store sign, with a mass of 20.0kg and 3.00m long, has its center of gravity at the center of the sign. It is supported by a loose bolt attached to the wall at one end and by a wire at the other end. The wire makes an angle of 25° with the horizontal. What is the tension in the wire?

AP ONLY: Oscillations

Simple Harmonic Motion (SHM) Oscillation: Simple Harmonic Motion (SHM) An object that “cycles” between having maximum kinetic and maximum potential energy.

Simple Harmonic Motion (SHM) - Notes This cycle will look like a sine or cosine wave The graph will look like this for displacement, potential energy, and kinetic energy as a function of time (It will just be shifted) 1 2 3 4

Simple Harmonic Motion (SHM) Determine these for our system: Amplitude = Maximum displacement from equilibrium Period (T) = Time for 1 complete motion Seconds per cycle Units = seconds Frequency = cycles per second Inverse of period Units = s-1 or Hertz (Hz)

Equations

Position Function for cosine – Class example If the object represented in the graph experiences a maximum displacement of 0.50m and completes a cycle in 2.0s, what is an appropriate expression for displacement as a function of time?

Position Function for cosine - Example A 0.50kg object on a spring (k=494) undergoes Simple Harmonic Motion. Its motion can be described by the following equation x = (0.3 m) cos (10π t) What is the amplitude of the vibration? What is the period of the vibration? What is the frequency? What is the maximum elastic PE? What is the maximum velocity?

AP ONLY: Gravitation

The force of gravity depends on 2 things: The mass of the two objects. Gravitation The force of gravity depends on 2 things: The mass of the two objects. The distance between them.

Gravitation - Example Mars has a mass 1/10 that of Earth and a diameter 1/2 that of Earth. The acceleration of a falling body near the surface of Mars is most nearly

Satellites Satellites in orbit are in centripetal motion. Fc is provided by the gravitational force. Combining Newton’s Law of Gravitation with the formula for centripetal force, we find: Rearrange for speed of satellite: Note that the mass of the satellite does not matter!

Example How long would it take a satellite to orbit the Mars if it were a distance of 6.79 x 106m from the center of the planet?

The force of gravity depends on 2 things: The mass of the two objects. Gravitation The force of gravity depends on 2 things: The mass of the two objects. The distance between them.

Gravitation - Example Mars has a mass 1/10 that of Earth and a diameter 1/2 that of Earth. The acceleration of a falling body near the surface of Mars is most nearly

Satellites Satellites in orbit are in centripetal motion. Fc is provided by the gravitational force. Combining Newton’s Law of Gravitation with the formula for centripetal force, we find: Rearrange for speed of satellite: Note that the mass of the satellite does not matter!

Example How long would it take a satellite to orbit the Mars if it were a distance of 6.79 x 106m from the center of the planet?