Lesson_3.1_Printable

Lesson_3.1_Printable - Work done by a force Work is done by...

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Work done by a force Work is done by a force when it produces a displacement Work done by a force is measured as the product of force and the displacement it produces in the direction of the force. force pplied on the box produces a displacement F x F θ Work done W = F N . x m = Fx Nm A force F applied on the box produces a displacement x on the box in the direction of the force Newton times meter is called a joule (J) In the diagram on the right, the force F applied on the box makes an angle θ with the direction of displacement. Here work is done by the component of F in the direction of displacement. 1
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Dot Product of two vectors If A and B are two vectors and the angle between them is θ , then the component of B in the direction of A is B cos θ . The product of A and the component of B in the direction of A is called the dot product of the two vectors nd is denoted by A B cos θ A B = || A || || || || B || Cos θ and is denoted by A B B cos θ A B = || || || || A || || || || || || B || || Cos θ os c A B A B θ = 1 cos A B A B - = 2
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A B B cos θ A B = || || A || || || || || || B || || Cos θ θ 1 cos A B A B θ - = If vectors A and B are represented in the component form, then the dot product of the two vectors is obtained by adding the product of the x components to the product of the y components. A = 2 i – 3 j B = 5 i + j A B = 10 – 3 = 7 The dot product of two vectors is a scalar . 3
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he component of the force in the F θ F cos θ x Consider a force F pulling a box at and angle θ with the horizontal The component of the force in the direction of displacement is F cos θ Work done W = (F cos θ ) N . (x )m = Fx cos θ J F cos θ produces a displacement x on the box in the direction of F cos θ The product of x vector and the component of F in the direction of x is the dot product of the vectors x and F . 4
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Work done is measure of the dot product of force and displacement vectors. F W = F · s Here s is the displacement vector θ F cos θ x F · s = (F)(s) cos θ θ is the angle between the two vectors. F s cos θ = (F)(s) The angle between the vectors F and s can be obtained as: 6
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Find A B and the angle between A and B if A = 4 i + 6 j and B = -2 i + 3 j. Also find the angle between A and B A B = -8 + 18 = 10 2 2 Magnitude of A = 4 6 + A B s = = 7.2 2 2 Magnitude of B = ( 2) 3 - + = 3.6 10 cos θ A B = 7.2 3.6 × = 0.386 θ = cos -1 (0.386) = 67 o 7
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A force vector F = 5 i – 2 j acting at a point produces a displacement given by s = 3 i – 4 j . Find the work done
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Lesson_3.1_Printable - Work done by a force Work is done by...

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