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Vectorsandscalars

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Kris Krianne Lionheart
Kris Krianne Lionheart

The document discusses scalar and vector quantities. It defines a scalar quantity as having only magnitude, and vector quantities as having both magnitude and direction. Length, time, and mass are provided as examples of scalars, while displacement, velocity, and force are examples of vectors. Vectors are represented using arrows with length indicating magnitude and direction indicating direction. The resultant of two vectors is found by combining them tail to tail using the parallelogram law or head to tail using the triangle law. Resolving vectors into perpendicular components and the relationships between magnitudes are also covered.

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Copyright 息 John OConnor St. Farnans PPS Prosperous For non-commercial purposes only.. Enjoy! Vectors and Scalars Comments/suggestions please to the SLSS physics website forum @ http://physics.slss.ie/forum
A scalar quantity is a quantity that has magnitude only and has no direction in space Examples of Scalar Quantities: Length Area Volume Time Mass
A vector quantity is a quantity that has both magnitude and a direction in space Examples of Vector Quantities: Displacement Velocity Acceleration Force
Vector diagrams are shown using an arrow The length of the arrow represents its magnitude The direction of the arrow shows its direction
Vectors in opposite directions: 6 m s-1 10 m s-1 = 4 m s-1 6 N 10 N = 4 N Vectors in the same direction: 6 N 4 N = 10 N 6 m = 10 m 4 m The resultant is the sum or the combined effect of two vector quantities
When two vectors are joined tail to tail Complete the parallelogram The resultant is found by drawing the diagonal When two vectors are joined head to tail Draw the resultant vector by completing the triangle
Solution: Complete the parallelogram (rectangle) 慮 The diagonal of the parallelogram ac represents the resultant force 2004 HL Section B Q5 (a) Two forces are applied to a body, as shown. What is the magnitude and direction of the resultant force acting on the body? 5N 12 N 5 12 a b c d The magnitude of the resultant is found using Pythagoras Theorem on the triangle abc N13 512Magnitude 22 = +== ac ac 属== = 67 5 12 tan 5 12 tan:ofDirection 1 慮 慮ac Resultant displacement is 13 N 67尊 with the 5 N force 13 N
45尊 5 N 90尊慮 Find the magnitude (correct to two decimal places) and direction of the resultant of the three forces shown below. 5N 5 5 Solution: Find the resultant of the two 5 N forces first (do right angles first) a b cd N07.75055 22 ==+=ac 属=== 451 5 5 tan 慮慮 7.07 N 10 N 135尊 Now find the resultant of the 10 N and 7.07 N forces The 2 forces are in a straight line (45尊 + 135尊 = 180尊) and in opposite directions So, Resultant = 10 N 7.07 N = 2.93 N in the direction of the 10 N force 2.93 N
What is a scalar quantity? Give 2 examples What is a vector quantity? Give 2 examples How are vectors represented? What is the resultant of 2 vector quantities? What is the triangle law? What is the parallelogram law?
When resolving a vector into components we are doing the opposite to finding the resultant We usually resolve a vector into components that are perpendicular to each other y v x Here a vector v is resolved into an x component and a y component
Here we see a table being pulled by a force of 50 N at a 30尊 angle to the horizontal When resolved we see that this is the same as pulling the table up with a force of 25 N and pulling it horizontally with a force of 43.3 N 50 Ny=25 N x=43.3 N 30尊 We can see that it would be more efficient to pull the table with a horizontal force of 50 N
If a vector of magnitude v and makes an angle 慮 with the horizontal then the magnitude of the components are: x = v Cos 慮 y = v Sin 慮 v y=v Sin 慮 x=v Cos 慮 慮 y Proof: v x Cos =慮 慮vCosx = v y Sin =慮 慮vSiny = x
60尊 2002 HL Sample Paper Section B Q5 (a) A force of 15 N acts on a box as shown. What is the horizontal component of the force? Vertical Component Horizontal Component Solution: N5.76015ComponentHorizontal =属== Cosx N99.126015ComponentVertical =属== Siny 15N 7.5 N 12.99N
A person in a wheelchair is moving up a ramp at constant speed. Their total weight is 900 N. The ramp makes an angle of 10尊 with the horizontal. Calculate the force required to keep the wheelchair moving at constant speed up the ramp. (You may ignore the effects of friction). Solution: If the wheelchair is moving at constant speed (no acceleration), then the force that moves it up the ramp must be the same as the component of its weight parallel to the ramp. 10尊 10尊80尊 900 N Complete the parallelogram. Component of weight parallel to ramp: N28.15610900 =属= Sin Component of weight perpendicular to ramp: N33.88610900 =属= Cos 156.28 N 886.33 N 2003 HL Section B Q6
If a vector of magnitude v has two perpendicular components x and y, and v makes and angle 慮 with the x component then the magnitude of the components are: x= v Cos 慮 y= v Sin 慮 v y=v Sin 慮 x=v Cos慮 慮 y

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Vectorsandscalars

  • 1. Copyright 息 John OConnor St. Farnans PPS Prosperous For non-commercial purposes only.. Enjoy! Vectors and Scalars Comments/suggestions please to the SLSS physics website forum @ http://physics.slss.ie/forum
  • 2. A scalar quantity is a quantity that has magnitude only and has no direction in space Examples of Scalar Quantities: Length Area Volume Time Mass
  • 3. A vector quantity is a quantity that has both magnitude and a direction in space Examples of Vector Quantities: Displacement Velocity Acceleration Force
  • 4. Vector diagrams are shown using an arrow The length of the arrow represents its magnitude The direction of the arrow shows its direction
  • 5. Vectors in opposite directions: 6 m s-1 10 m s-1 = 4 m s-1 6 N 10 N = 4 N Vectors in the same direction: 6 N 4 N = 10 N 6 m = 10 m 4 m The resultant is the sum or the combined effect of two vector quantities
  • 6. When two vectors are joined tail to tail Complete the parallelogram The resultant is found by drawing the diagonal When two vectors are joined head to tail Draw the resultant vector by completing the triangle
  • 7. Solution: Complete the parallelogram (rectangle) 慮 The diagonal of the parallelogram ac represents the resultant force 2004 HL Section B Q5 (a) Two forces are applied to a body, as shown. What is the magnitude and direction of the resultant force acting on the body? 5N 12 N 5 12 a b c d The magnitude of the resultant is found using Pythagoras Theorem on the triangle abc N13 512Magnitude 22 = +== ac ac 属== = 67 5 12 tan 5 12 tan:ofDirection 1 慮 慮ac Resultant displacement is 13 N 67尊 with the 5 N force 13 N
  • 8. 45尊 5 N 90尊慮 Find the magnitude (correct to two decimal places) and direction of the resultant of the three forces shown below. 5N 5 5 Solution: Find the resultant of the two 5 N forces first (do right angles first) a b cd N07.75055 22 ==+=ac 属=== 451 5 5 tan 慮慮 7.07 N 10 N 135尊 Now find the resultant of the 10 N and 7.07 N forces The 2 forces are in a straight line (45尊 + 135尊 = 180尊) and in opposite directions So, Resultant = 10 N 7.07 N = 2.93 N in the direction of the 10 N force 2.93 N
  • 9. What is a scalar quantity? Give 2 examples What is a vector quantity? Give 2 examples How are vectors represented? What is the resultant of 2 vector quantities? What is the triangle law? What is the parallelogram law?
  • 10. When resolving a vector into components we are doing the opposite to finding the resultant We usually resolve a vector into components that are perpendicular to each other y v x Here a vector v is resolved into an x component and a y component
  • 11. Here we see a table being pulled by a force of 50 N at a 30尊 angle to the horizontal When resolved we see that this is the same as pulling the table up with a force of 25 N and pulling it horizontally with a force of 43.3 N 50 Ny=25 N x=43.3 N 30尊 We can see that it would be more efficient to pull the table with a horizontal force of 50 N
  • 12. If a vector of magnitude v and makes an angle 慮 with the horizontal then the magnitude of the components are: x = v Cos 慮 y = v Sin 慮 v y=v Sin 慮 x=v Cos 慮 慮 y Proof: v x Cos =慮 慮vCosx = v y Sin =慮 慮vSiny = x
  • 13. 60尊 2002 HL Sample Paper Section B Q5 (a) A force of 15 N acts on a box as shown. What is the horizontal component of the force? Vertical Component Horizontal Component Solution: N5.76015ComponentHorizontal =属== Cosx N99.126015ComponentVertical =属== Siny 15N 7.5 N 12.99N
  • 14. A person in a wheelchair is moving up a ramp at constant speed. Their total weight is 900 N. The ramp makes an angle of 10尊 with the horizontal. Calculate the force required to keep the wheelchair moving at constant speed up the ramp. (You may ignore the effects of friction). Solution: If the wheelchair is moving at constant speed (no acceleration), then the force that moves it up the ramp must be the same as the component of its weight parallel to the ramp. 10尊 10尊80尊 900 N Complete the parallelogram. Component of weight parallel to ramp: N28.15610900 =属= Sin Component of weight perpendicular to ramp: N33.88610900 =属= Cos 156.28 N 886.33 N 2003 HL Section B Q6
  • 15. If a vector of magnitude v has two perpendicular components x and y, and v makes and angle 慮 with the x component then the magnitude of the components are: x= v Cos 慮 y= v Sin 慮 v y=v Sin 慮 x=v Cos慮 慮 y