Problem 1
What is the percent uncertainty in the measurement 3.67 ± 0.25 m? (Ans: 6.8%)
Given Data:
- Measured value (x) = 3.67 m
- Absolute uncertainty (Δx) = 0.25 m
- Percent Uncertainty = ?
Solution:
The formula for percent uncertainty is:
Substituting the given values:
Result:
The percentage uncertainty in length is 6.81%.
Problem 2
What is the area, and its approximate uncertainty, of a circle with radius 3.7 × 104 cm? (Ans: 4.3 × 109 cm2 ± 5.4%)
Given Data:
- Radius (r) = 3.7 × 104 cm
- Least count/uncertainty in radius (Δr) = 0.1 × 104 cm
- Area (A) = ?
- Uncertainty in Area (ΔA) = ?
Solution:
First, let's calculate the area of the circle using the formula:
For a power function like r2, the percentage uncertainty in area is twice the percentage uncertainty in the radius:
Now, find the absolute uncertainty in Area (ΔA):
Result:
The area of the circle is 4.30 × 109 cm2 and its absolute uncertainty is 0.23 × 109 cm2 (or 5.4%).
Problem 3
An aeroplane travels at 850 km/h. How long does it take to travel 1.00 km? (Ans: 4.23 s)
Given Data:
- Speed of aeroplane (v) = 850 km/h
- Distance traveled (S) = 1.00 km
- Time taken (t) = ?
Solution:
According to the standard definition of uniform speed:
Substituting the values to get time in hours:
Converting hours into seconds (1 hour = 3600 seconds):
Result:
It will take 4.24 seconds for the aeroplane to travel a distance of 1.00 km.
Problem 4
A rectangular holding tank 25.0 m in length and 15.0 m in width is used to store water for a short period of time in an industrial plant. If 2980 m3 of water is pumped into the tank, what is the depth of the water? (Ans: 7.95 m)
Given Data:
- Length of the tank (l) = 25.0 m
- Width of the tank (w) = 15.0 m
- Volume of water (V) = 2980 m3
- Depth of water (h) = ?
Solution:
The total volume of a rectangular prism layout is given by:
Rearranging the formula to isolate the depth variable (h):
Substituting the given configuration details:
Result:
The depth of the water inside the rectangular holding tank is 7.95 m.
Problem 5
Find the volume of a rectangular underground water tank that has storage facilities dimensions of 1.9 m by 1.2 m by 0.8 m. (Ans: 1.824 m3)
Given Data:
- Length (l) = 1.9 m
- Width (w) = 1.2 m
- Depth / Height (h) = 0.8 m
- Volume (V) = ?
Solution:
The total volume formula for a rectangular prism container shape is:
Substituting the matrix dimensions:
Result:
The volume of the rectangular tank is 1.824 m3.
Problem 6
Two students derive following equations in which x refers to distance traveled, v the speed, a the acceleration, and t the time (the subscript 'o' means a quantity at time t = 0):
(a) x = vt2 + 2at
(b) x = vot + 2at2
Which of these could possibly be correct according to a dimensional check?
(Ans: Equation b is dimensional correct)
Standard Dimensions:
- Distance [x] = [L]
- Velocity [v] = [LT-1]
- Acceleration [a] = [LT-2]
- Time [t] = [T]
Solution Analysis:
Checking Equation (a): x = vt2 + 2atDimension of Left Hand Side (L.H.S):
Dimension of Right Hand Side (R.H.S):
Comparing Equation (i) and (ii), [L.H.S] ≠ [R.H.S]. Thus, equation (a) is incorrect.
Checking Equation (b): x = vot + 2at2Dimension of Left Hand Side (L.H.S):
Dimension of Right Hand Side (R.H.S):
Comparing Equation (iii) and (iv), [L.H.S] = [R.H.S]. Thus, equation (b) is correct.
Result:
Student (a)'s derived equation is dimensionally incorrect and Student (b)'s derived equation is dimensionally correct.
Problem 7
One hectare is defined as 104 m2. One acre is 4 × 104 ft2. How many acres are in one hectare? (Hint: 1 m = 3.28 ft) (Ans: 2.69 acres)
Given Data:
- 1 Hectare = 104 m2
- 1 Acre = 4 × 104 ft2
- Conversion factor: 1 m = 3.28 ft ⇒ 1 m2 = (3.28)2 ft2 = 10.7584 ft2
Solution:
Step 1: Convert the value of 1 Hectare from m2 into ft2:
Step 2: Find total number of acres inside one hectare by dividing their area layouts:
Result:
There are approximately 2.69 acres in one hectare.
Problem 8
A watch factory claims that its watches gain or lose not more than 10 seconds in a year. How accurate is this watch, expressed as a percentage? (Ans: 3.17 × 10-5 %)
Given Data:
- Error / Uncertainty (Δt) = 10 s
- Total Time period (t) = 1 year = 365 days × 24 hrs × 60 mins × 60 s = 31,536,000 s
- Percentage Error / Accuracy = ?
Solution:
Result:
The variance accuracy error percentage in this watch is 3.17 × 10-5%.
Problem 9
The diameter of the Moon is 3480 km. What is the volume of the Moon? How many Moons would be needed to create a volume equal to that of Earth? (Hint: Radius of Earth = 6380 km) (Ans: Vmoon = 2.21 × 1010 km3, 49 Moons)
Given Data:
- Diameter of Moon (Dm) = 3480 km
- Radius of Moon (Rm) = Dm / 2 = 1740 km
- Radius of Earth (Re) = 6380 km
- Volume of Moon (Vm) = ?
- Number of Moons needed (N) = ?
Solution:
Both celestial bodies are spherical. The volume of a sphere is given by:
Result:
The volume of the Moon is 2.21 × 1010 km3 and approximately 49 Moons would be needed to match the volume of the Earth.
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