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lecturenotes-Jan25

lecturenotes-Jan25 - Example and are vectors Vectors and...

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, , and, are vectors. Vectors and when added together equal the vector . Vector has a magnitude of 88 units and it is directed at an angle of 44 ° relative to the x axis as shown. Find the scalar components of vectors and . B x B y C x C y a) 63 0 0 61 b) 0 61 63 0 c) 63 0 61 0 d) 0 63 0 61 e) 61 0 63 0 Example 61 63
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Vectors: magnitude and direction Adding/subtracting vectors: Graphical and components Multiplying vectors: 1) Vector*scalar = Vector 2) Vector*vector = scalar (Scalar Product or Dot Product) 3) Vector*vector = Vector (Cross Product) Quick Review: Vectors A B = A B cos θ = A x B x + A y B y + A z B z C = A × B C = A B sin θ = area of parallelogram B A θ C
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Quick Review: 2D Kinematics 2-D with CONSTANT ACCELERATION The x and y motions are decoupled : this means that we can consider both directions of motion independently. x –direction motion y –direction motion v x = v ox + a x t x = 1 2 v x + v o x ( ) t x = v ox t + 1 2 a x t 2 v x 2 = v ox 2 + 2 a x x v y = v oy + a y t y = 1 2 v y + v o y ( ) t y = v oy t + 1 2 a y t 2 v y 2 = v oy 2 + 2 a y y Δ r = Δ x ˆ i + Δ y ˆ j v = v x ˆ i + v y ˆ j a = a x ˆ i + a y ˆ j
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Quick Review: Projectile Motion Toss something in the air: a x = 0 and a y = -g x –direction motion y –direction motion horizontal vertical v x = v ox x = v ox t v y = v oy + a y t y = 1 2 v y + v o y ( ) t y = v oy t + 1 2 a y t 2 v y 2 = v oy 2 + 2 a y y
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A Real Trajectory
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Uniform circular motion: Constant speed in circular path Centripetal Acceleration “center-seeking” - Always directed inwards ! - Always constant magnitude ? a c = v 2 r Period - time it takes an object to travel circumference of circle T = 2 π r v
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A steel ball is whirled on the end of a chain in a horizontal circle of radius R with a constant period T . If the radius of the circle is then reduced to 0.75 R , while the period remains T , what happens to the centripetal acceleration of the ball? a) The centripetal acceleration increases to 1.33 times its initial value. b) The centripetal acceleration increases to 1.78 times its initial value. c) The centripetal acceleration decreases to 0.75 times its initial value. d) The centripetal acceleration decreases to 0.56 times its initial value. e) The centripetal acceleration does not change. Example
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Example What is the magnitude of the centripetal acceleration of an object on the Earth’s equator owing to the rotation of the Earth? a c = v 2 r a c = v 2 r = 2 π r T 2 r = 4 π 2 r T 2 = 4 π 2 6.37 × 10 6 m ( ) 1 day ( ) 86400 s 1 day 2 = 0.033 m / s 2 = 0.3% g DIRECTED INWARD just like gravity
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Relative motion in 1-D Velocity of passenger (P) depends on which reference frame your in. What is the man in the train’s velocity with respect to a bird flying overhead to the left with a speed of +20 m/s (wrt ground)?
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