Physics 11th Chap 1 : Topic 2 - Maryam Publications

Physics Chapter 1 : Topic 2

Dimensional Homogeneity

1 / 35

A physical equation is said to be dimensionally homogeneous if:

A) All terms contain the same unit prefix.

B) All quantities are measured in $\mathbf{SI}$ units.

C) Dimensions on both sides of the equation are identical.

D) Numerical values on both sides are equal.

2 / 35

Which of the following equations is dimensionally correct?

A) $\mathbf{v = u +}\mathbf{at}^{\mathbf{2}}$

B) $\mathbf{v = u + at}$

C) $\mathbf{v = ut + a}$

D) $\mathbf{v = u + a\ /\ t}$

3 / 35

The equation $\mathbf{s = ut +}$ $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{at}^{\mathbf{2}}$ is dimensionally correct because:

A) All terms have dimensions of acceleration.

B) Each term has dimensions of length.

C) All terms are dimensionless.

D) Constants make it correct.

4 / 35

Which equation is dimensionally incorrect?

A) $\mathbf{p = mv}$

B) $\mathbf{F = ma}$

C) $\mathbf{P =}$ $\frac{\mathbf{F}}{\mathbf{A}}$

D) $\mathbf{E = mv}$

5 / 35

The dimensions of $\mathbf{v}^{\mathbf{2}}$ are:

A) $\left\lbrack \mathbf{L\ }\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

B) $\left\lbrack \mathbf{L\ }\mathbf{T}^{\mathbf{- 1}} \right\rbrack$

C) $\left\lbrack \mathbf{L}^{\mathbf{2\ }}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

D) $\left\lbrack \mathbf{L}^{\mathbf{2\ }}\mathbf{T}^{\mathbf{- 1}} \right\rbrack$

6 / 35

Which equation passes the dimensional test?

A) $\mathbf{v = u + as}$

B) $\mathbf{v}^{\mathbf{2}}\mathbf{=}\mathbf{u}^{\mathbf{2}}\mathbf{+ 2as}$

C) $\mathbf{v =}\mathbf{u}^{\mathbf{2}}\mathbf{+ as}$

D) $\mathbf{v}^{\mathbf{2}}\mathbf{= u + 2as}$

7 / 35

In dimensional analysis, numerical constants such as $\mathbf{2}$, $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\ }$, and $\mathbf{\pi}$ are:

A) Ignored because they are dimensionless

B) Converted into $\mathbf{SI}$ units

C) Treated as having dimensions

D) Added to dimensions

8 / 35

Which quantity has dimensions $\left\lbrack \mathbf{L}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$?

A) Velocity

B) Force

C) Momentum

D) Acceleration

9 / 35

If $\mathbf{F =}\mathbf{mv}^{\mathbf{2}}$, the equation is:

A) Correct for all systems

B) Numerically correct only

C) Dimensionally incorrect

D) Dimensionally homogeneous

10 / 35

Which equation is dimensionally valid?

A) $\mathbf{s = vt + a}$

B) $\mathbf{s = v}\mathbf{t}^{\mathbf{2}}\mathbf{+ a}$

C) $\mathbf{s = vt +}$ $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{a}\mathbf{t}^{\mathbf{2}}$

D) $\mathbf{s = v + at}$

11 / 35

The dimensions of force are:

A) $\left\lbrack \mathbf{M\ L\ }\mathbf{T}^{\mathbf{- 1}} \right\rbrack$

B) $\left\lbrack \mathbf{M\ }\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

C) $\left\lbrack \mathbf{M\ L\ }\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

D) $\left\lbrack \mathbf{M\ }\mathbf{L}^{\mathbf{2\ }}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

12 / 35

Which equation is dimensionally incorrect?

A) $\mathbf{(P = W\ /\ t)}$

B) $\mathbf{(P = W + t)}$

C) $\mathbf{(E = Pt)}$

D) $\mathbf{(W = Fs)}$

13 / 35

If dimensions of LHS and RHS are different, then:

A) Equation may still be physically correct.

B) Equation is definitely incorrect.

C) Constants can fix it.

D) Equation becomes dimensionless.

14 / 35

Which quantity has dimensions $\left\lbrack \mathbf{M\ }\mathbf{L}^{\mathbf{2\ }}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$?

A) Energy

B) Pressure

C) Force

D) Momentum

15 / 35

The equation $\mathbf{T = 2\pi}$ $\sqrt{\frac{\mathbf{l}}{\mathbf{g}}}$ is dimensionally correct because:

A) $\mathbf{l}$ and $\mathbf{g}$ have same dimensions.

B) $\mathbf{\pi}$ has dimensions of time.

C) $\frac{\mathbf{l}}{\mathbf{g}}$ has dimensions $\mathbf{T}^{\mathbf{2}}$.

D) $\mathbf{2\pi}$ balances dimensions.

16 / 35

The dimensions of pressure are:

A) $\left\lbrack \mathbf{M\ }\mathbf{L}^{\mathbf{- 2\ }}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

B) $\left\lbrack \mathbf{M\ }\mathbf{L}^{\mathbf{2\ }}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

C) $\left\lbrack \mathbf{M\ L\ }\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

D) $\left\lbrack \mathbf{M\ }\mathbf{L}^{\mathbf{- 1\ }}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

17 / 35

Which equation is dimensionally homogeneous?

A) $\mathbf{F = mv + a}$

B) $\mathbf{F = ma + t}$

C) $\mathbf{F = ma}$

D) $\mathbf{F = m + a}$

18 / 35

The dimensions of momentum are:

A) $\left\lbrack \mathbf{M\ }\mathbf{L}^{\mathbf{2\ }}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

B) $\left\lbrack \mathbf{M\ L\ }\mathbf{T}^{\mathbf{- 1}} \right\rbrack$

C) $\left\lbrack \mathbf{M\ L\ }\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

D) $\left\lbrack \mathbf{M\ }\mathbf{T}^{\mathbf{- 1}} \right\rbrack$

19 / 35

Which statement about dimensional analysis is correct?

A) It determines numerical constants.

B) It can prove an equation is correct in every respect.

C) It replaces experiments.

D) It can detect dimensional errors in equations.

20 / 35

The dimensions of $\mathbf{a}^{\mathbf{2\ }}\mathbf{t}^{\mathbf{2}}$ are:

A) $\left\lbrack \mathbf{L\ }\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

B) $\left\lbrack \mathbf{L\ }\mathbf{T}^{\mathbf{- 1}} \right\rbrack$

C) $\left\lbrack \mathbf{L}^{\mathbf{2\ }}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

D) $\left\lbrack \mathbf{L}^{\mathbf{2}} \right\rbrack$

21 / 35

An equation passes dimensional analysis. This means:

A) Numerical constants are correct.

B) It is certainly correct.

C) Experimental verification is unnecessary.

D) It may be correct dimensionally.

22 / 35

Which equation is dimensionally wrong?

A) $\mathbf{P =}$ $\frac{\mathbf{W}}{\mathbf{t}}$

B) $\mathbf{KE =}$ $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{m}\mathbf{v}^{\mathbf{2}}$

C) $\mathbf{PE = mgh}$

D) $\mathbf{F = mv}$

23 / 35

Dimensions of gravitational acceleration ‘$\mathbf{g}$’ are:

A) $\left\lbrack \mathbf{L\ }\mathbf{T}^{\mathbf{- 1}} \right\rbrack$

B) $\left\lbrack \mathbf{L\ }\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

C) $\left\lbrack \mathbf{M\ }\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

D) $\left\lbrack \mathbf{L}^{\mathbf{2\ }}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$

24 / 35

Which quantity has dimensions $\left\lbrack \mathbf{M\ }\mathbf{L}^{\mathbf{- 1\ }}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$?

A) Energy

B) Density

C) Force

D) Pressure

25 / 35

If $\mathbf{E =}$ $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{mv}$, then the equation is:

A) Homogeneous

B) Dimensionless

C) Numerically correct

D) Dimensionally incorrect

26 / 35

The dimensions of $\mathbf{P \times t\ (power \times time)}$ are:

A) Force

B) Momentum

C) Pressure

D) Energy

27 / 35

Which equation is dimensionally valid?

A) $\mathbf{v =}\mathbf{u}^{\mathbf{2}}\mathbf{+ at}$

B) $\mathbf{v = u +}\mathbf{at}^{\mathbf{2}}$

C) $\mathbf{v = u + at}$

D) $\mathbf{v}^{\mathbf{2}}\mathbf{= u + at}$

28 / 35

The dimensions of density are:

A) $\left\lbrack \mathbf{M\ }\mathbf{L}^{\mathbf{- 1}} \right\rbrack$

B) $\left\lbrack \mathbf{M\ }\mathbf{L}^{\mathbf{- 3}} \right\rbrack$

C) $\left\lbrack \mathbf{M}^{\mathbf{- 1\ }}\mathbf{L}^{\mathbf{3}} \right\rbrack$

D) $\left\lbrack \mathbf{M\ }\mathbf{L}^{\mathbf{- 2}} \right\rbrack$

29 / 35

Which equation correctly relates dimensions?

A) $\mathbf{Energy = Pressure \times Area}$

B) $\mathbf{Energy = Force \times Distance}$

C) $\mathbf{Energy = Force \div Distance}$

D) $\mathbf{Energy = Momentum \times}\mathbf{Velocity}^{\mathbf{- 1}}$

30 / 35

Which of the following is a limitation of dimensional analysis?

A) It cannot determine numerical constants like $\mathbf{2\ }$or $\mathbf{\pi}$.

B) It cannot check homogeneity.

C) It cannot compare dimensions.

D) It cannot identify dimensions of quantities.

31 / 35

The equation $\mathbf{v = u + 2at}$ is dimensionally:

A) Dimensionless

B) Incorrect because velocity and acceleration differ

C) Incorrect because of $\mathbf{2}$

D) Correct because $\mathbf{2}$ is dimensionless

32 / 35

Which equation is dimensionally homogeneous but physically incorrect?

A) $\mathbf{F = ma}$

B) $\mathbf{P =}$ $\frac{\mathbf{W}}{\mathbf{t}}$

C) $\mathbf{s = ut + a}\mathbf{t}^{\mathbf{2}}$ (for a specific derivation, coefficient missing)

D) $\mathbf{s = ut +}$ $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{a}\mathbf{t}^{\mathbf{2}}$

33 / 35

Why is dimensional homogeneity important?

A) It eliminates measurement errors.

B) It converts $\mathbf{SI}$ units automatically.

C) It guarantees experimental accuracy.

D) It helps verify whether an equation can possibly be correct.

34 / 35

Which quantity has dimensions $\left\lbrack \mathbf{L}^{\mathbf{2\ }}\mathbf{T}^{\mathbf{- 2}} \right\rbrack$?

A) Acceleration

B) Force

C) Velocity

D) Kinetic energy per unit mass

35 / 35

If an equation contains terms with dimensions of length, length, and velocity, the equation is:

A) Not dimensionally homogeneous

B) Correct

C) Dimensionless

D) Homogeneous

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