General Physics

Let's examine each pair:
Option (A) **Galileo Galilei - Law of inertia:** Galileo Galilei is credited with the initial formulation of the principle of inertia, although Isaac Newton later formalized it into his first law of motion.
So, this pair is largely correct.

Option (B) **Michael Faraday - Law of electromagnetic induction:** Michael Faraday is indeed credited with the discovery of the law of electromagnetic induction, a fundamental principle in electromagnetism.
So, this pair is correct.

Option (C) **Rudolf Hertz - Generation of electromagnetic waves:** Rudolf Hertz is famous for being the first to conclusively demonstrate the existence of electromagnetic waves, predicted by James Clerk Maxwell's equations.
So, this pair is correct.

Option (D) **C.
V.
Raman - Wave theory of light:** C.
V.
Raman is renowned for his work on the scattering of light, which led to the discovery of the Raman effect.
While his work contributed significantly to the understanding of light-matter interaction, the wave theory of light was primarily established by scientists like Christiaan Huygens and Thomas Young, much earlier than Raman's work.
Therefore, the wrongly matched pair is C.
V.
Raman - Wave theory of light.

Step 1: Dimensional Analysis 1. Work and Torque - Work: - Torque: - Same dimensional formula. 2. Angular Momentum and Planck’s Constant - Angular Momentum: - Planck’s Constant: - Same dimensional formula. 3. Stress and Linear Momentum - Stress: - Linear Momentum: - Different dimensional formula. 4. Surface Tension and Force Constant - Surface Tension: - Force Constant: - Same dimensional formula. Conclusion: Stress and linear momentum do not share the same dimensional formula. Thus, the correct answer is:

First, perform the subtraction: Aligning decimal places: 0.
312 - 0.
030 ------- 0.
282 In subtraction, the result is reported to the same number of decimal places as the number with the fewest decimal places.
0.
312 has 3 decimal places.
0.
03 has 2 decimal places.
So, the result of subtraction should have 2 decimal places.
.
If we were to report this intermediate step based on sig figs from subtraction rules, it would be 0.
28 (last significant digit is the hundredths place, same as 0.
03).
However, it's generally better to keep extra digits during intermediate calculations and round at the end.
Let's use 0.
282 for now.
Number of significant figures in 0.
282 is 3.
Now the expression is .
Consider the number of significant figures in each term: 0.
501 has 3 significant figures.
0.
05 has 1 significant figure (the 5; leading zeros are not significant).
0.
282 has 3 significant figures.
For multiplication and division, the result should have the same number of significant figures as the measurement with the fewest significant figures.
In this case, 0.
05 has only 1 significant figure.
Therefore, the final result of the entire calculation should be reported with 1 significant figure.
Let's calculate the value: Now, .
If we strictly followed sig figs at each step: (2 sig figs, as per subtraction rule for decimal places, but 0.
03 has 1 sig fig in value, the last digit is significant).
The rule for subtraction/addition: result has same number of decimal places as the number with least decimal places.
0.
312 (3 d.
p) 0.
03 (2 d.
p) → result to 2 d.
p.
→ 0.
28.
(2 sig figs) Then .
The term with fewest sig figs is 0.
05 (1 sig fig).
So the final answer should have 1 significant figure.
Let's calculate without intermediate rounding: Value = Rounding this to 1 significant figure gives 3.
Number of significant figures in "3" is 1.
If the rules are applied more loosely, focusing on the precision of the numbers: 0.
501 (precision to 0.
001) 0.
05 (precision to 0.
01) 0.
312 (precision to 0.
001) 0.
03 (precision to 0.
01) .
Given 0.
03 has its last sig fig at 2nd decimal place, .
Retain as 0.
28 for sig fig rules of subtraction based on decimal places.
Number of sig figs in 0.
501 is 3.
Number of sig figs in 0.
05 is 1.
Number of sig figs in 0.
28 (from 0.
282 based on subtraction rule) is 2.
The limiting term for multiplication/division is 0.
05 with 1 significant figure.
So the final answer should have 1 significant figure.
The value is .
Rounded to 1 sig fig is 3.
However, the provided answer is 2 significant figures (option 3).
This implies that perhaps 0.
05 is considered to have 2 significant figures (e.
g.
, if it was written as 0.
050, or if it's an exact number).
If 0.
05 is exact or has more sig figs implicitly.
If 0.
05 has 2 sig figs (e.
g.
, treated as 0.
050), then the terms are: 0.
501 (3 sig figs) 0.
050 (2 sig figs) .
The subtraction .
Since 0.
030 has its last sig fig at the 3rd decimal place, the result is good to 3rd decimal place.
Or using the original 0.
03 (2 decimal places), then rounded to 2 decimal places is .
(2 sig figs).
Using 0.
501 (3sf), 0.
050 (2sf), 0.
28 (2sf).
The minimum is 2 sf.
So the result rounded to 2 significant figures is .
The number of significant figures is 2.
This matches option (3).
This outcome depends critically on interpreting 0.
05 as having two significant figures (e.
g.
, if it was measured as 0.
050 or is an exact divisor).
Standard interpretation of "0.
05" is one significant figure.
If the numbers are from measurements: 0.
501 (3 s.
f.
) 0.
05 (1 s.
f.
) 0.
312 (3 s.
f.
) 0.
03 (1 s.
f.
) Subtraction: .
Last sig.
digit of 0.
03 is at 100ths place.
Last sig.
digit of 0.
312 is at 1000ths place.
Result to 100ths place: .
(2 s.
f.
) Then .
The result should have 1 s.
f.
Value , rounded to 1 s.
f.
is 3.
Number of s.
f is 1.
If we take 0.
03 as having 2 sig figs (e.
g.
0.
030), then (3sf).
Then .
Result to 1sf.
The only way to get 2 significant figures as the answer is if 0.
05 is assumed to have at least 2 significant figures, and the result of subtraction is used with 3 sig figs (or more precisely, from , having 2 sig figs).
If 0.
05 has 2 sig figs (e.
g.
0.
050), and (2 sig figs).
Then .
The result should have 2 sig figs.
Calculated value is .
Rounded to 2 sig figs is .
Number of sig figs = 2.
This path leads to the answer.

Step 1: Determine the dimensions of Young's modulus
Young's modulus is stress divided by strain. Stress = force/area, dimensions: . Strain is dimensionless. Thus, Young's modulus has dimensions . Step 2: Compare with the options
% Option (1) Strain: dimensionless, .
% Option (2) Gravitational potential: energy/mass, .
% Option (3) Surface energy: energy/area, .
% Option (4) Energy density: energy/volume, , matches Young's modulus. Step 3: Select the correct option
Energy density has the same dimensions as Young's modulus, so the answer is (4).

The relationship between Fahrenheit ( ) and Celsius ( ) is given by the linear equation .
This is a linear relationship, so the graph of versus is a straight line with a slope of and a y-intercept of 32.

Step 1: Differentiate displacement to get velocity. Step 2: Find critical points (minimum/maximum velocity). Set derivative of velocity to zero: Step 3: Find velocity at Step 4: Confirm it's minimum by second derivative test. % Final Answer

First, perform the subtraction: Aligning decimal places: 0.
312 - 0.
030 ------- 0.
282 In subtraction, the result is reported to the same number of decimal places as the number with the fewest decimal places.
0.
312 has 3 decimal places.
0.
03 has 2 decimal places.
So, the result of subtraction should have 2 decimal places.
.
If we were to report this intermediate step based on sig figs from subtraction rules, it would be 0.
28 (last significant digit is the hundredths place, same as 0.
03).
However, it's generally better to keep extra digits during intermediate calculations and round at the end.
Let's use 0.
282 for now.
Number of significant figures in 0.
282 is 3.
Now the expression is .
Consider the number of significant figures in each term: 0.
501 has 3 significant figures.
0.
05 has 1 significant figure (the 5; leading zeros are not significant).
0.
282 has 3 significant figures.
For multiplication and division, the result should have the same number of significant figures as the measurement with the fewest significant figures.
In this case, 0.
05 has only 1 significant figure.
Therefore, the final result of the entire calculation should be reported with 1 significant figure.
Let's calculate the value: Now, .
If we strictly followed sig figs at each step: (2 sig figs, as per subtraction rule for decimal places, but 0.
03 has 1 sig fig in value, the last digit is significant).
The rule for subtraction/addition: result has same number of decimal places as the number with least decimal places.
0.
312 (3 d.
p) 0.
03 (2 d.
p) → result to 2 d.
p.
→ 0.
28.
(2 sig figs) Then .
The term with fewest sig figs is 0.
05 (1 sig fig).
So the final answer should have 1 significant figure.
Let's calculate without intermediate rounding: Value = Rounding this to 1 significant figure gives 3.
Number of significant figures in "3" is 1.
If the rules are applied more loosely, focusing on the precision of the numbers: 0.
501 (precision to 0.
001) 0.
05 (precision to 0.
01) 0.
312 (precision to 0.
001) 0.
03 (precision to 0.
01) .
Given 0.
03 has its last sig fig at 2nd decimal place, .
Retain as 0.
28 for sig fig rules of subtraction based on decimal places.
Number of sig figs in 0.
501 is 3.
Number of sig figs in 0.
05 is 1.
Number of sig figs in 0.
28 (from 0.
282 based on subtraction rule) is 2.
The limiting term for multiplication/division is 0.
05 with 1 significant figure.
So the final answer should have 1 significant figure.
The value is .
Rounded to 1 sig fig is 3.
However, the provided answer is 2 significant figures (option 3).
This implies that perhaps 0.
05 is considered to have 2 significant figures (e.
g.
, if it was written as 0.
050, or if it's an exact number).
If 0.
05 is exact or has more sig figs implicitly.
If 0.
05 has 2 sig figs (e.
g.
, treated as 0.
050), then the terms are: 0.
501 (3 sig figs) 0.
050 (2 sig figs) .
The subtraction .
Since 0.
030 has its last sig fig at the 3rd decimal place, the result is good to 3rd decimal place.
Or using the original 0.
03 (2 decimal places), then rounded to 2 decimal places is .
(2 sig figs).
Using 0.
501 (3sf), 0.
050 (2sf), 0.
28 (2sf).
The minimum is 2 sf.
So the result rounded to 2 significant figures is .
The number of significant figures is 2.
This matches option (3).
This outcome depends critically on interpreting 0.
05 as having two significant figures (e.
g.
, if it was measured as 0.
050 or is an exact divisor).
Standard interpretation of "0.
05" is one significant figure.
If the numbers are from measurements: 0.
501 (3 s.
f.
) 0.
05 (1 s.
f.
) 0.
312 (3 s.
f.
) 0.
03 (1 s.
f.
) Subtraction: .
Last sig.
digit of 0.
03 is at 100ths place.
Last sig.
digit of 0.
312 is at 1000ths place.
Result to 100ths place: .
(2 s.
f.
) Then .
The result should have 1 s.
f.
Value , rounded to 1 s.
f.
is 3.
Number of s.
f is 1.
If we take 0.
03 as having 2 sig figs (e.
g.
0.
030), then (3sf).
Then .
Result to 1sf.
The only way to get 2 significant figures as the answer is if 0.
05 is assumed to have at least 2 significant figures, and the result of subtraction is used with 3 sig figs (or more precisely, from , having 2 sig figs).
If 0.
05 has 2 sig figs (e.
g.
0.
050), and (2 sig figs).
Then .
The result should have 2 sig figs.
Calculated value is .
Rounded to 2 sig figs is .
Number of sig figs = 2.
This path leads to the answer.

Since the source is approaching, we use the relativistic Doppler effect formula for light: Where: Now, substitute into the formula: Convert to km/s: