An object is released from rest and falls in free fall motion. The speed v of the object after it has fallen a distance y is given by
v2 = 2gy.
In an experiment, v and y are measured and the measured values are used to calculate g. If the percent uncertainty in the measured value of v is 4.13% and the percent uncertainty in the measured value of y is 5.00%, what is the percent uncertainty in the calculated value of g? (Do not enter units for this answer.)
7.645
To find the percent uncertainty in the calculated value of g, we need to use the formula for percent uncertainty.
The percent uncertainty is given by:
Percent uncertainty = (Uncertainty / Measured value) * 100
Let's calculate the percent uncertainty in the calculated value of g step-by-step.
Step 1: Calculate the percent uncertainty in the measured value of v.
Percent uncertainty in v = 4.13%
Step 2: Calculate the percent uncertainty in the measured value of y.
Percent uncertainty in y = 5.00%
Step 3: Calculate the percent uncertainty in the calculated value of g using the formula for propagation of uncertainties.
Percent uncertainty in g = sqrt((2 * Percent uncertainty in y)^2 + (Percent uncertainty in v)^2)
Step 4: Substitute the given values into the formula.
Percent uncertainty in g = sqrt((2 * 5.00%)^2 + (4.13%)^2)
Step 5: Evaluate the expression.
Percent uncertainty in g ≈ sqrt((10.00%)^2 + (4.13%)^2)
Percent uncertainty in g ≈ sqrt(0.1000^2 + 0.0413^2)
Percent uncertainty in g ≈ sqrt(0.0100 + 0.00171)
Percent uncertainty in g ≈ sqrt(0.01171)
Percent uncertainty in g ≈ 0.1082
Step 6: Convert to a percentage.
Percent uncertainty in g ≈ 10.82%
Therefore, the percent uncertainty in the calculated value of g is approximately 10.82%.
To find the percent uncertainty in the calculated value of g, we need to determine how the uncertainties in the measured values of v and y affect the calculated value of g.
Let's start by analyzing how the uncertainties in v and y affect the uncertainty in the calculated value of g:
Given:
- The equation to calculate v² is v² = 2gy.
- The percent uncertainty in the measured value of v is 4.13%.
- The percent uncertainty in the measured value of y is 5.00%.
To find the percent uncertainty in g, we can use the concept of relative uncertainty:
Relative uncertainty = (absolute uncertainty) / (measured value)
First, let's calculate the absolute uncertainty in v:
Absolute uncertainty in v = (4.13% of v) = (4.13/100) * v.
Second, let's calculate the absolute uncertainty in y:
Absolute uncertainty in y = (5.00% of y) = (5.00/100) * y.
Now, let's substitute the uncertainties in the equation v² = 2gy and solve for g:
(v + Δv)² = 2g(y + Δy)
Expanding the equation:
v² + 2vΔv + Δv² = 2gy + 2gΔy
Since we're interested in the value of g, we can ignore the second-order term Δv²:
v² + 2vΔv = 2gy + 2gΔy
Subtracting v² from both sides of the equation:
2vΔv = 2gy + 2gΔy - v²
Factoring out g:
2vΔv = 2g(y + Δy) - v²
Dividing both sides of the equation by 2v:
Δv/v = (g(y + Δy) - v²) / (v(y + Δy))
The left-hand side of the equation represents the percent uncertainty in v, and the right-hand side represents the ratio of uncertainties in g, y, and v.
Now, the percent uncertainty in v is given as 4.13%:
(Δv/v) * 100 = 4.13%
Substituting the expression for Δv/v derived above:
[ (g(y + Δy) - v²) / (v(y + Δy)) ] * 100 = 4.13%
Now, let's solve for Δg/g, the relative uncertainty in g:
Δg/g = (g(y + Δy) - v²) / (v(y + Δy))
Rearranging the equation:
g(y + Δy) - v² = (v(y + Δy)) * (Δg/g)
Expanding the terms on the left-hand side of the equation:
gy + gΔy - v² = (v(y + Δy)) * (Δg/g)
Dividing both sides of the equation by gy:
1 + (Δg/g) * (Δy/y) - (v² / gy) = (v * (Δy/y + 1)) * (Δg/gy)
Here, the term on the left-hand side of the equation can be ignored since v² / gy is negligible compared to 1.
Simplifying further:
(Δg/g) = (v * (Δy/y + 1)) * (Δg/gy)
Now, let's substitute the percent uncertainties:
(Δg/g) = (v * (0.05 + 1)) * (0.0413)
Now we can calculate the percent uncertainty in the calculated value of g:
(Δg/g) * 100 = (v * 1.05 * 0.0413) * 100
Therefore, the percent uncertainty in the calculated value of g is approximately 4.33%.