d1 = 2.53 cm +/- .05 cm
d2 = 1.753 m +/- .001 m
0 = 23.5 degrees +/- .5 degrees
v1 = 1.55 m/s +/- .15 m/s
Using the measured quantities above, calculate the following. Express the uncertainty calculated value.
Z = 4d1 (cos (0))^ 2
To calculate the value of Z, we need to substitute the given values of d1 and 0 into the equation Z = 4d1 (cos(0))^2. Let's break down the calculation step-by-step:
Step 1: Calculate the value of cos(0)
- We can use a calculator to find the cosine of 23.5 degrees, which is our given angle.
- cos(23.5 degrees) ≈ 0.919
Step 2: Square the cosine value
- (cos(23.5 degrees))^2 ≈ 0.919^2 ≈ 0.844
Step 3: Calculate 4d1
- Multiply the value of d1 by 4: 4 * 2.53 cm ≈ 10.12 cm
Step 4: Multiply Step 3 and Step 2 to get Z
- Multiply 10.12 cm by 0.844: 10.12 cm * 0.844 ≈ 8.54 cm
Now, let's consider the uncertainty in our result, Z:
To determine the uncertainty in Z, we follow the general rule of error propagation, which states that for multiplication or division, the relative error (uncertainty) adds up.
Relative error = (absolute uncertainty) / (measured value)
Step 1: Calculate the relative uncertainty for d1:
- Relative uncertainty for d1 = (0.05 cm) / (2.53 cm) ≈ 0.0198
Step 2: Calculate the relative uncertainty for (cos(0))^2:
- Since the angle uncertainty is given as a relative uncertainty, it remains the same for cos(0).
- Relative uncertainty for (cos(0))^2 = 0.5 degrees / 23.5 degrees ≈ 0.0213
Step 3: Calculate the combined relative uncertainty for 4d1 (cos(0))^2:
- Combine the relative uncertainties by summing them: 0.0198 + 0.0213 ≈ 0.0411
Step 4: Calculate the uncertainty in Z by multiplying the combined relative uncertainty by the calculated value of Z:
- Uncertainty in Z = 0.0411 * 8.54 cm ≈ 0.35 cm
Therefore, the final result is:
Z = 8.54 cm +/- 0.35 cm