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Units And Measurement

Question
CBSEENPH11017695
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A particle of mass m is projected from the ground with velocity u at angle 0 with horizontal. Find the angular momentum of the particle about the point of projection at any instant. Deduce the result when a particle will be at the highest point. 

Solution
Consider the point of projection as origin, the horizontal direction as X-axis and vertically upward direction positive direction as Y-axis.

The space velocity space of space particle space at space any space instant space straight t space is comma space straight v with rightwards harpoon with barb upwards on top space equals space straight u space cos space straight theta space straight i with hat on top space plus space left parenthesis space straight u space sinθ space minus space straight t right parenthesis thin space straight j with hat on top The space position space of space the space particle space at space any space instant space is comma space straight r with rightwards harpoon with barb upwards on top space equals space left parenthesis space straight u space cos space θt right parenthesis space straight i with hat on top space minus space left parenthesis space straight u space sinθt space minus space 1 half gt squared right parenthesis space straight j with hat on top space Now comma space the space angular space momentum space of space the space particle space is comma space straight L with rightwards harpoon with barb upwards on top space equals space straight r with rightwards harpoon with barb upwards on top space straight x space straight p with rightwards harpoon with barb upwards on top space equals space straight m space straight r with rightwards harpoon with barb upwards on top straight x space straight v with rightwards harpoon with barb upwards on top space space space equals space straight m open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row ucosθt cell straight u space sinθt space minus space 1 half gt squared end cell cell space 0 space end cell row ucosθ cell usinθ space minus space gt end cell 0 end table close vertical bar space space space space equals space minus space 1 half mug space cosθ space straight t squared space straight k with hat on top space Time space at space which space the space particle space will space be space at space the space highest space point comma space straight t space equals space fraction numerator straight u space sin space straight theta over denominator straight g end fraction
Therefore, angular momentum of particle when it will be at highest point is,
L with rightwards harpoon with barb upwards on top space equals negative 1 half m u g space space c o s space 0 space straight t squared space space straight k with hat on top space space space equals space minus 1 half m u g space c o s space straight theta space open parentheses fraction numerator straight u space s i n space straight theta over denominator straight g end fraction close parentheses squared space straight k with hat on top space space space space equals space minus space fraction numerator m u cubed c o s space theta space s i n squared 0 over denominator 2 straight g end fraction straight k with hat on top

Some More Questions From Units and Measurement Chapter

What are the two main required characteristics of a unit?

Can we use same units for different physical quantities?

Can we use different units to measure same physical quantity?

To measure a physical quantity X, the unit U is repeatedly used for n times. What is the magnitude of physical quantity?

Does the numerical factor in the quantitative measurement depend on the system of unit used?

Is one unit sufficient to measure a physical quantity?

Can a physical quantity have two or more units? Give example.

What are fundamental quantities?

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