# Numericals » 01-Numerical Problems

## “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 :

1. Young’s Modulus.
2. Coefficient of Viscosity.
3. Planck’s constant.
4. Boltzmann’s Constant.
5. Coefficient of thermal conductivity.
6. Joule’s mechanical equivalent of heat.

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1. Dimensional Analysis.

### 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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1. Dimensional Analysis.

### 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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1. Dimensional Analysis.

### 04) PROBLEM – P010104

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

1. T = 2 \pi \sqrt { \frac {l}{g}}
2. \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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1. Dimensional Analysis.

### 05) PROBLEM – P010105

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

1. its length ( l ) .
2. its mass per unit length ( m ) .
3. the tension ( T ) in the string.

Obtain dimensionally an expression for frequency ( \nu ) .

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1. Dimensional Analysis.

### 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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