## “Basics of Science” Numerical Problems

Try to solve the following “Basics of Science” numerical problems to clear the concepts in solving the numerical problems.

First of all go to the theory portion of the respective topic and then try to solve the numerical problems by yourself. If facing problem in solving the numerical, click on the *Go to solution * button to see the ready made solution placed at the bottom of each numerical problem.

__01) PROBLEM – P010101__

*Deduce the dimensional formulae for the following physical quantities :*

*Young’s Modulus.**Coefficient of Viscosity.**Planck’s constant.**Boltzmann’s Constant.**Coefficient of thermal conductivity.**Joule’s mechanical equivalent of heat.*

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__02) PROBLEM – P010102__

*In CGS system, the value of Stefan’s constant is ( \sigma = 5.67 \times 10^{-5} \text {erg}^{-1} \text {cm}^{-2} K^{-4} ) **. Find the value in SI units. Given ( 1 j = 10^{-7} \text {erg} ) *

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__03) PROBLEM – P010103__

*If the unit of force is ( 1 \ kN ) **, unit of length ( 1 \ km ) ** and the unit of time is ( 100 \ s ) **, what will be the unit of mass?*

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__04) PROBLEM – P010104__

*Check by the method of dimensions, whether the following equations are correct:*

- T = 2 \pi \sqrt { \frac {l}{g}}
- \nu = \frac {1}{2l} \sqrt { \frac {l}{m}}
*where ( \nu ) =**frequency of vibration, ( l ) =**length of the string, ( T ) =**tension in the string.*

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__05) PROBLEM – P010105__

*The frequency ( \nu ) = ** of vibration of a stretched string depends upon –*

*its length ( l )**.**its mass per unit length ( m )**.**the tension ( T )**in the string.*

* Obtain dimensionally an expression for frequency ( \nu ) **.*

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__06) PROBLEM – P010106__

* ABCD is a parallelogram and ( \overrightarrow {AC} ) ** and ( \overrightarrow {BD} ) ** are its diagonals. Prove that –*

*(1)** \overrightarrow {AC} + \overrightarrow {BD} = 2 \overrightarrow {BC} \quad and (2) \overrightarrow {AC} - \overrightarrow {BD} = 2 \overrightarrow {AB} *

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__07) PROBLEM – P010107__

*A particle has a displacement of** ( 12 \ m ) towards east and ( 5 \ m )** towards the north and then ( 6 \ m ) ** vertically upward. Find the magnitude of the sum of these displacements.*

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__08) PROBLEM – P010108__

*A boatman can row with a speed of ( 10 \ km h^{-1} ) ** in still water. If the river flows steadily at ( 5 \ km h^{-1} ) **, in which direction should the boatman row in order to reach a point on the other bank directly opposite to the point from where he started? The width of the river is ( 2 \ km ) .*

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__09) PROBLEM – P010109__

*A vector ( \vec {X} ) **, when added to the resultant of the vectors ( \vec {A} = 3 \hat i - 5 \hat j + 7 \hat k ) ** and ( \vec {B} = 2 \hat i + 4 \hat j - 3 \hat k ) ** gives a unit vector along Y-axis. Find the vector ( \vec {X} ) **.*

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__10) PROBLEM – P010110__

*If vector ( \vec {A} = 3 \hat i + 4 \hat j ) **and ( \vec {A} = 7 \hat i + 24 \hat j ) ** **then find a vector having the same magnitude as ( \vec {B} ) **and parallel to vector ( \vec {A} ) **.*

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__11) PROBLEM – P010111__

*If | \vec {A} + \vec {B} | = | \vec {A} - \vec {B} | , **then find the angle between vectors ( \vec {A} ) \text {and} ( \vec {B} ) .** *

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__12) PROBLEM – P010112__

*Find the angle between the vectors ( \vec {A} = \hat i + 2 \hat j - \hat k ) **and ( \vec {A} = - \hat i + \hat j - 2 \hat k ) **.*

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__13) PROBLEM – P010113__

*Find a vector whose length is 7 ** and which is perpendicular to each of the vectors ( \vec {A} = 2 \hat i - 3 \hat j + 6 \hat k ) ** and ( \vec {B} = \hat i + \hat j - \hat k ) .*

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__14) PROBLEM – P010114__

*The diagonals of a parallelogram are given by the vectors ( 2 \hat i + \hat j + 2 \hat k ) ** **and ( \hat i - 3 \hat j + 4 \hat k ) .** **Find the area of the parallelogram.*

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