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Lesson 3.1 Work

# Lesson 3.1 Work - Work done by a force Work is done by a...

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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. 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) F x F 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. θ 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 . A B = | | A | | | | 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 and is denoted by A B A B B cos θ A B = || A || || B || Cos θ θ os c A B A B θ = 1 cos A B A B θ - = 2
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 A B B cos θ A B = || A || || B || Cos θ θ 1 cos A B A B θ - = B = 5 i + j A B = 10 – 3 = 7 The dot product of two vectors is a scalar . 3

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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 . F θ F cos θ x Consider a force F pulling a box at and angle θ with the horizontal 4
5

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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
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 cos θ = A B = 7.2 2 2 Magnitude of B = ( 2) 3 - + = 3.6 10 = 7.2 3.6 × = 0.386 = 67 o 7

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