© Shannon W. Helzer. All Rights Reserved. 1 Unit 6 Analytical Vector Addition

© Shannon W. Helzer. All Rights Reserved. 2 Multiplying a Vector by a Scalar A A C = -1/2 A B = 2A C B A ½ A

© Shannon W. Helzer. All Rights Reserved. 3 Adding “-” Vectors C = A + B D = A - B D = A + (- B) A B C -B D  Add “negative” vectors by keeping the same magnitude but adding 180 degrees to the direction of the original vector.

© Shannon W. Helzer. All Rights Reserved. 4 Components of Vectors A = A x + A y AyAy AxAx Recall: Vectors are always added “head to tail.” A 

© Shannon W. Helzer. All Rights Reserved. 5 Components of Vectors A = A x + A y Finding the components when you know A. Recall:  is measured from the positive x axis.

© Shannon W. Helzer. All Rights Reserved. 6 If A = 3.0 m and  = 45 , find A x + A y. AyAy AxAx A   From the + x Axis

© Shannon W. Helzer. All Rights Reserved. 7 Components of Vectors Finding the vector magnitude and direction when you know the components. Recall:  is measured from the positive x axis. Caution: Beware of the tangent function. Always consider in which quadrant the vector lies when dealing with the tangent function.

© Shannon W. Helzer. All Rights Reserved. 8 If A x = 2.0 m and A y. = 2.0 m, then Find A and . AyAy AxAx A  

© Shannon W. Helzer. All Rights Reserved. 9 Adding Vectors With Components.

© Shannon W. Helzer. All Rights Reserved. 10 Follow this type of methodology when doing these problems. MagnitudeAngleRxRy    A=72.4m m61.4 m B = 57.3 m m m C = 17.8 m m m Rx = mRy = 9.92m R = 12.7 mAngle = 129 degrees Component Template

© Shannon W. Helzer. All Rights Reserved. 11 Hand Glider Trip

© Shannon W. Helzer. All Rights Reserved. 12 Analytical Vector Addition – Hand Glider VectorMagnitudeAnglex componenty componentQuad  Find the Final displacement of the hand glider using analytical vector addition.  This problem is similar to the following problems: WS 21, 1; WS 22, 1; and WS 23, 4.

© Shannon W. Helzer. All Rights Reserved. 13 Analytical Vector Addition VectorMagnitudeAnglex componenty componentQuad  Three Physics track robots pull on a book as shown.  They pull with the following forces: , , and .  Find the net force applied to this most valuable book.  This problem is similar to WS 21, 2.

© Shannon W. Helzer. All Rights Reserved. 14 Statics – Hanging Sign  Draw the Static FBD for the sign below  What do we need to do with the tension (T)?  Resolve it into its components (T x & T y ).

© Shannon W. Helzer. All Rights Reserved. 15 Statics – Hanging Sign  Draw the Static FBD for the sign below  What do we need to do with the tension (T)?  Resolve it into its components (Tx & Ty).

© Shannon W. Helzer. All Rights Reserved. 16 Inclined Plane Problems  Draw the FBD for the piano on the inclined plane.  What will we have to do with the Normal Force (N) and the force of friction (Ff)?  Resolve them into their x and y components.

© Shannon W. Helzer. All Rights Reserved. 17 Inclined Plane Problems  Would you like to do less work?  How could we do this problem by resolving only one force?  Try rotating the FBD so that the N is in the y plane and the Ff is in the x plane.

© Shannon W. Helzer. All Rights Reserved. 18 Analytical Vector Addition VectorMagnitudeAnglex componenty componentQuad F1F1 F2F2 F3F3 F4F F  Use the table below when performing analytical vector addition.  Do WS 23 numbers 1 & 2.

© Shannon W. Helzer. All Rights Reserved. 19 VrVr Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 20 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 21 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 22 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 23 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 24 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 25 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 26 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 27 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 28 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 29 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 30 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 31 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 32 Analytical Vector Addition VectorMagnitudeAnglex componenty componentQuad F1F1 F2F2 F3F3 F4F F  Do WS 23 number 3.  This problem is similar to WS 24, 3 and WS 22, 2.

© Shannon W. Helzer. All Rights Reserved. 33 This presentation was brought to you by Where we are committed to Excellence In Mathematics And Science Educational Services.

© Shannon W. Helzer. All Rights Reserved. 34 Analytical Vector Addition VectorMagnitudeAnglex componenty componentQuad F1F1 F2F2 F3F3 F4F F AA

© Shannon W. Helzer. All Rights Reserved. 35 Setting the Standard  When we do problems involving kinematics, it is important that we stick to a standard when imputing data into the know- want table.  This standard enables us to take into account the vector nature of acceleration, velocity, displacement, etc.  Here is a diagram we will use in order to help us correctly input data into the table.  This standard is based upon the Cartesian Coordinate system.  If a body travels West, then what sign would you give its velocity?  If a body travels at an angle of 90 degrees, then what sign would you give its velocity? 3-3

© Shannon W. Helzer. All Rights Reserved. 36 Dot Products  Write each of the three vectors given in their unit vector notation.  A =   B =   C =   Calculate the Dot Products below. 3-3

© Shannon W. Helzer. All Rights Reserved. 37 Dot Products – Finding the angle  Given the vectors below, find the angles between the following vectors.  A and C.  B and A.  C and E.  D and E. 3-3

© Shannon W. Helzer. All Rights Reserved D Cartisian Coordinate System +x +z +y  1-17

© Shannon W. Helzer. All Rights Reserved. 39 Unit Vectors -i -j j k i -k z y x A = A x i + A y j + A z k Note: Remember to put the “^” over the hand written vector when writing unit vectors. 1-18

© Shannon W. Helzer. All Rights Reserved. 40 Scalar or “Dot” Product B A B A  The Dot product gives the projection of one vector onto another. You can also use the dot product to find the angle between the vectors. B A = Projection of B onto A. BABA A B = Projection of A onto B. ABAB 1-19

© Shannon W. Helzer. All Rights Reserved. 41 Scalar or “Dot” Product B A  The Dot product results in a Scalar quantity. 1-20

© Shannon W. Helzer. All Rights Reserved. 42 Scalar or “Dot” Product & Unit Vectors You “multiply” the dot product in a similar way as below. A = A x i + A y jB = B x i + B y j AB = (A x i + A y j) ( B x i + B y j) AB = A x i B x i + A x i B y j+ A y j B x i + A y j B y j However, i i = j j = k k = 1 i j = i k = j k = 0 AB = A x B x + A y B y 1-21

© Shannon W. Helzer. All Rights Reserved. 43 Scalar or “Dot” Product B A  One use for the dot product is to determine the angle between two vectors. 1-22

© Shannon W. Helzer. All Rights Reserved. 44 Vector or “Cross” Product   Right hand rule: Place the fingers of your right hand in the direction of the first vector in the cross product. Rotate your fingers towards the second vector. Your thumb tells you the direction of the resultant vector. The Cross product results in a VECTOR quantity. 1-23

© Shannon W. Helzer. All Rights Reserved. 45 Vector or “Cross” Product  The Cross product results in a VECTOR quantity. The magnitude of the vector is given by WARNING: AB sin  DOES NOT EQUAL BA sin  A x B DOES NOT EQUAL B x A However, A x B = - B x A 1-24

© Shannon W. Helzer. All Rights Reserved. 46 Vector or “Cross” Product AxB = (A x i + A y j) x ( B x i + B y j) AxB = A x i x B x i + A x i x B y j+ A y j x B x i + A y j x B y j However, i x i = j x j = k x k = 0 i x j = -j x i = k j x k = -k x j = i K x i = -i x k = j AxB = A x i x B y j+ A y j x B x i AxB = (A x B y ) i x j + ( A y B x ) j x i AxB = (A x B y ) k - ( A y B x ) k 1-25

© Shannon W. Helzer. All Rights Reserved. 47 Vector or “Cross” Product A = A x i + A y j + A z kB = B x i + B y j + B z k A x B = (A x i + A y j + A z k) x ( B x i + B y j + B z k) A x B = A y B z i - A z B y i+ A z B x j- A x B z j+ A x B y k - A y B x k A x B = (A y B z –A z B y )i + (A z B x -A x B z )j + (A x B y -A y B x )k Determinant method of solving for the cross product. A x B = Rx i + Ry j + Rz k 1-26

© Shannon W. Helzer. All Rights Reserved. 48 Spherical Coordinates

© Shannon W. Helzer. All Rights Reserved. 49 Advanced Physics Unit 5 Applications of Newton’s Laws

© Shannon W. Helzer. All Rights Reserved. 50 Newton’s Laws – A Review  Newton’s First Law - An object remains at rest, or in uniform motion in a straight line, unless it is compelled to change by an externally imposed force.  Newton’s first law describes an Equilibrium Situation.  An Equilibrium Situation is one in which the acceleration of a body is equal to zero.  Newton’s Second Law – If there is a non-zero net force on a body, then it will accelerate.  Newton’s Second Law describes a Non-equilibrium Situation.  A Non-equilibrium Situation is one in which the acceleration of a body is not equal to zero.  Newton’s Third Law - for every action force there is an equal, but opposite, reaction force.

© Shannon W. Helzer. All Rights Reserved. 51 Free Body Diagrams – A Review  When solving problems involving forces, we must draw FBDs of all bodies involved in the force interactions.  Since torque is related to force, we must modify the FBD concept to apply to bodies upon which a torque acts.  Before we carry out this modification, lets review problems involving force using FBDs.  If the crate started from rest, then which way did it accelerate?  Draw the FBD for the crate.  What type of a situation is depicted below? Dig Dug

© Shannon W. Helzer. All Rights Reserved. 52 Free Body Diagrams – Inclined Plane (WS 14 # 8)  A block slides down an inclined plane as shown.  Draw the FBD for the block as it slides down the ramp at a constant speed.  Write the Newton’s laws in vector form for the block in both the horizontal and vertical directions.  Now convert from vector form to math form.

© Shannon W. Helzer. All Rights Reserved. 53 WS 14 Problems 1-4  A mass rest on an inclined plane as shown.  What type of friction is acting on the mass?  Now suppose the mass begins to slide down the plane.  What type of friction is acting on the mass as it slides?  Draw the FBD for the mass while at rest and while sliding down the plane.

© Shannon W. Helzer. All Rights Reserved. 54 Static Friction v. Kinetic Friction  Static friction exists when an object wants to move but is held in place by the force of friction.  This force of friction is greater than the component of the weight acting down the plane.  If we continue to rotate the plane, the component of the weight acting down the plane will eventually become larger than the normal force.  When this happens, the object will begin to slide changing from static friction to kinetic friction.

© Shannon W. Helzer. All Rights Reserved. 55 Like WS 14 Problem 5  MeanyBot and PhysicsBot are moving a crate as shown.  MeanyBot is pulling with a force F 2 = 10,000 N and the PhysicsBot is pushing with a force of F 1 = 6,000 N.  Additionally, the coefficient of kinetic friction,  k, is  The mass of the crate is 1000 kg.  Determine the net force and the acceleration of the crate.

© Shannon W. Helzer. All Rights Reserved. 56 WS 14 Problem 9  Derive the equations needed to determine the tension and the acceleration of the weights (m 1 <m 2 ) on the Atwood’s machine shown to the right..  What type of a situation do we have when the masses first begin to move?  Define this situation with its two predominate characteristics.

© Shannon W. Helzer. All Rights Reserved. 57

© Shannon W. Helzer. All Rights Reserved. 58 WS 14 Problem 10 – Elevator Problems  An elevator (m = kg) ascending at a rate of 8.5 m/s comes to a stop in a distance of 22.0 m.  Find the Tension in the three cables supporting the weight of the elevator and the acceleration experienced by the elevator.  What type of a situation is the elevator in while coming to a stop?  Now suppose Dr. Physics (m = 62.5 kg) is standing on a scale inside the elevator.  After three seconds of descending, the elevator begins traveling at a constant speed of 9.0 m/s.  What does the scale say that Dr. Physics weighs while he descends?  What situation is the elevator in once it begins traveling at a constant velocity?

© Shannon W. Helzer. All Rights Reserved. 59 WS 15 Problem 1  Block A below weighs 90.0 N.  The coefficient of static friction between the block and the table is  s =  Block B weighs 15.0 N.  The system is in equilibrium.  Draw and label the FBDs for Body A & Body B.  What is meant by the term “equilibrium” above?  Find the friction force acting on Block A. B 45° A

© Shannon W. Helzer. All Rights Reserved. 60 WS 15 Problem 2  A wooden block (m 1 ) rests on a plane inclined at an angle of .  This block is attached to mass m 2 held at a height of y above the ground.  The coefficient of friction between the block and the incline is  K.  Derive the equations (in terms of m 1, m 2,  K, y, , and g) needed to calculate the tension in the string, the acceleration of the system, and the time needed for to hit the ground.

© Shannon W. Helzer. All Rights Reserved. 61 WS 15 Problem 3  Derive the equations needed to determine the tension in each chain given the angle , that the angle between chain 2 and the post is 90 , and the fact that the weight of the bug zapper is W.

© Shannon W. Helzer. All Rights Reserved. 62 WS 15 Problem 4  Derive the equations needed to determine m 2 and the tensions in the strings given angles  and  that the mass of weight one is m 1.  Suppose m 1 = 10.0 kg. What are the tensions and what is the value of m 2 ?  Suppose m 1 = 6.0 kg. What are the tensions and what is the value of m 2 ?  Suppose m 1 = 2.9 kg. What are the tensions and what is the value of m 2 ?

© Shannon W. Helzer. All Rights Reserved  A wrapped box (m 1 ) rests on a table and is attached to a hanging weight (m 2 ) as shown.  The coefficient of friction between the box and the table is  K.  The weight is released pulling the box to the right as shown.  Derive the equations (in terms of m 1, m 2,  K, and g) needed to calculate the tension in the string and the acceleration of the system.

© Shannon W. Helzer. All Rights Reserved. 64 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 65 Relative Velocity

© Shannon W. Helzer. All Rights Reserved. 66 Torque aa

© Shannon W. Helzer. All Rights Reserved. 67 Torque aa

© Shannon W. Helzer. All Rights Reserved. 68 Torque aa

© Shannon W. Helzer. All Rights Reserved. 69 Static Friction v. Kinetic Friction  Static friction exist when an object wants to move but is held in place by the force of friction.  This force of friction is greater than the component of the weight acting down the plane.  If we continue to rotate the plane, the component of the weight acting down the plane will eventually become larger than the normal force.  When this happens, the object will begin to slide changing from static friction to kinetic friction.

© Shannon W. Helzer. All Rights Reserved. 70 Torque aa

© Shannon W. Helzer. All Rights Reserved. 71 Torque aa

© Shannon W. Helzer. All Rights Reserved. 72 Torque aa

© Shannon W. Helzer. All Rights Reserved. 73 Advanced Physics Unit 5 Exam QUESTION QUESTION 4 QUESTION 5 QUESTION 6

© Shannon W. Helzer. All Rights Reserved. 74 Unit 5 Exam Problems 1-3  The graph below shows the force of friction verses the pull time.  What type of friction is represented in the portion of the graph that is blue in color?  What type of friction is represented in the portion of the graph that is red in color?  What is physically happening at the point where the graph changes from blue to red? RETURN

© Shannon W. Helzer. All Rights Reserved. 75 Unit 5 Exam Problem 4  Under what conditions would a Elevator passenger appear to weigh more than his or her actual weight: while accelerating upwards, while accelerating downwards, or while riding at a constant speed?  Justify your answer using verbal explanations or equations as needed. RETURN RIDE

© Shannon W. Helzer. All Rights Reserved. 76 Unit 5 Exam Problem 5  A wrapped box (m 1 ) rests on a table and is attached to a hanging weight (m 2 ) as shown.  The coefficient of friction between the box and the table is  K.  The weight is released pulling the box to the right as shown.  Derive the equations (in terms of m 1, m 2,  K, and g) needed to calculate the tension in the string and the acceleration of the system. RETURN

© Shannon W. Helzer. All Rights Reserved. 77 Unit 5 Exam Problem 6  Derive the equations needed to determine the tension in each chain given angles  and  and the fact that the weight of the bug zapper is W. RETURN