Chapter 1, Problem 36P

(a)

To determine

To Write: The dimension of the constants $C_1$ and $C_2$ .

Answer to Problem 36P

The dimension of the constant $C_1$ is $\left[\text{L}\right]$ and $C_2$ is $\left[\text{L}\right]\left[{\text{T}}^{-1}\right]$ .

Explanation of Solution

Introduction:

Dimension of any physical quantity in physics tells about how any quantity is related to the fundamental quantities of mass, length and time. Dimensions of any quantity is determined by its definition.

Dimension of mass is $\left[\text{M}\right]$ , dimension of length is $\left[\text{L}\right]$ , and dimension of time is $\left[\text{T}\right]$ .

Unit can be defined as the definite amount of quantity used for standard of measurement. The standard of length is metre and similarly for time is second and for mass is kilogram.

Any two quantities having different dimensions can be multiplied but they cannot be added or subtracted. Dimension of two quantities must remain same during addition or subtraction but not during multiplication,

Write the expression for distance.

  $x=C_1+C_{2}t$

Here, $x$ is the distance, $t$ is time, $C_1$ and $C_2$ are constants.

The dimension of x is $\left[\text{L}\right]$ ; the dimension of $C_1$ is the unit of x. Thus, the dimension of $C_1$ is $\left[\text{L}\right]$ .

The dimension of t is $\left[\text{T}\right]$ ; the dimension of $C_2$ is the dimension of $x/t$ . Thus, the unit of $C_2$ is $\left[\text{L}\right]\left[{\text{T}}^{-1}\right]$ .

(b)

To determine

To Write: The dimension of the constant $C_1$ .

Answer to Problem 36P

The dimension of the constant $C_1$ is $\left[\text{L}\right]\left[{\text{T}}^{-2}\right]$ .

Explanation of Solution

Introduction:

Dimension of any physical quantity in physics tells about how any quantity is related to the fundamental quantities of mass, length and time. Dimensions of any quantity is determined by its definition.

Dimension of mass is $\left[\text{M}\right]$ , dimension of length is $\left[\text{L}\right]$ , and dimension of time is $\left[\text{T}\right]$ .

Unit can be defined as the definite amount of quantity used for standard of measurement. The standard of length is metre and similarly for time is second and for mass is kilogram.

Any two quantities having different dimensions can be multiplied but they cannot be added or subtracted. Dimension of two quantities must remain same during addition or subtraction but not during multiplication,

Write the expression for distance.

  $x=\frac{1}{2}{C}_1{t}^2$

Here, $x$ is the distance, $t$ is time and $C_1$ is constant.

The dimension of x is $\left[\text{L}\right]$ ; the dimension of t is $\left[\text{T}\right]$ ; the dimension of $C_1$ is the unit of $x/t^2$ . Thus, the dimension of $C_1$ is $\left[\text{L}\right]\left[{\text{T}}^{-2}\right]$ .

(c)

To determine

To Write: The dimension of the constant $C_1$ .

Answer to Problem 36P

The dimension of the constant $C_1$ is $\left[\text{L}\right]\left[{\text{T}}^{-2}\right]$ .

Explanation of Solution

Introduction:

Dimension of any physical quantity in physics tells about how any quantity is related to the fundamental quantities of mass, length and time. Dimensions of any quantity is determined by its definition.

Dimension of mass is $\left[\text{M}\right]$ , dimension of length is $\left[\text{L}\right]$ , and dimension of time is $\left[\text{T}\right]$ .

Unit can be defined as the definite amount of quantity used for standard of measurement. The standard of length is metre and similarly for time is second and for mass is kilogram.

Any two quantities having different dimensions can be multiplied but they cannot be added or subtracted. Dimension of two quantities must remain same during addition or subtraction but not during multiplication,

Write the expression for distance.

  $v^2=2C_{1}x$

Here, $x$ is the distance, $v$ is velocity and $C_1$ is constant.

The dimension of x is $\left[\text{L}\right]$ ; The dimension of v is $\left[\text{L}\right]\left[{\text{T}}^{-1}\right]$ ; the dimension of $C_1$ is the dimension of $v^2/x$ . Thus, the dimension of $C_1$ is $\left[\text{L}\right]\left[{\text{T}}^{-2}\right]$ .

(d)

To determine

To Write: The dimension of the constants $C_1$ and $C_2$ .

Answer to Problem 36P

The dimension of $C_1$ is $\left[\text{L}\right]$ and $C_2$ is $\left[{\text{T}}^{-1}\right]$ .

Explanation of Solution

Introduction:

Dimension of any physical quantity in physics tells about how any quantity is related to the fundamental quantities of mass, length and time. Dimensions of any quantity is determined by its definition.

Dimension of mass is $\left[\text{M}\right]$ , dimension of length is $\left[\text{L}\right]$ , and dimension of time is $\left[\text{T}\right]$ .

Unit can be defined as the definite amount of quantity used for standard of measurement. The standard of length is metre and similarly for time is second and for mass is kilogram.

Any two quantities having different dimensions can be multiplied but they cannot be added or subtracted. Dimension of two quantities must remain same during addition or subtraction but not during multiplication,

Write the expression for distance.

  $x=C_1\cos C_{2}t$

Here, $x$ is the distance, $t$ is time, $C_1$ and $C_2$ are constants.

The dimension of x is $\left[\text{L}\right]$ ; the dimension of $C_1$ is the unit of x. Thus, the dimension of $C_1$ is $\left[\text{L}\right]$ .

The dimension of t is $\left[\text{T}\right]$ ; the dimension of $C_2$ is the dimension of $1/t$ . Thus, the dimension of $C_2$ is $\left[{\text{T}}^{-1}\right]$ .

(e)

To determine

To Write: The dimension of the constants $C_1$ and $C_2$ .

Answer to Problem 36P

The dimension of the constant $C_1$ is $\left[\text{L}\right]\left[{\text{T}}^{-1}\right]$ and $C_2$ is $\left[{\text{T}}^{-1}\right]$ .

Explanation of Solution

Introduction:

Dimension of any physical quantity in physics tells about how any quantity is related to the fundamental quantities of mass, length and time. Dimensions of any quantity is determined by its definition.

Dimension of mass is $\left[\text{M}\right]$ , dimension of length is $\left[\text{L}\right]$ , and dimension of time is $\left[\text{T}\right]$ .

Unit can be defined as the definite amount of quantity used for standard of measurement. The standard of length is metre and similarly for time is second and for mass is kilogram.

Any two quantities having different dimensions can be multiplied but they cannot be added or subtracted. Dimension of two quantities must remain same during addition or subtraction but not during multiplication,

Write the expression for distance.

  $v^2=2C_{1}v-{\left(C_{2}x\right)}^2$

Here, $x$ is the distance, $v$ is velocity, $C_1$ and $C_2$ are constants.

The dimension of v is $\left[\text{L}\right]\left[{\text{T}}^{-1}\right]$ ; the dimension of $C_1$ is the unit of $v^2/v$ . Thus, the dimension of the constant $C_1$ is $\left[\text{L}\right]\left[{\text{T}}^{-1}\right]$ .

The dimension of x is $\left[\text{L}\right]$ ; the dimension of v is $\left[\text{L}\right]\left[{\text{T}}^{-1}\right]$ ; the dimension of $C_2$ is the dimension of $v/x$ . Thus, the unit of $C_2$ is $\left[{\text{T}}^{-1}\right]$ .