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.
d1 (tan (0))
To calculate the value of d1 multiplied by the tangent of 0, we first need to find the value of tan(0) and then multiply it by d1.
Step 1: Calculate the value of tan(0)
The given angle is 0 = 23.5 degrees +/- 0.5 degrees.
To calculate tan(0) with uncertainty, we can use the tangent function and propagate the uncertainties using the formula:
delta_tan(0) = | d(tan(0))/d(0) | * delta(0)
Here, delta_tan(0) represents the uncertainty in the tangent, d(tan(0))/d(0) is the derivative of the tangent function with respect to the angle, and delta(0) is the uncertainty in the angle.
The derivative of the tangent function with respect to the angle is sec^2(0).
So, delta_tan(0) = sec^2(0) * delta(0)
To calculate delta_tan(0), we substitute the given angle and its uncertainty:
delta_tan(0) = sec^2(23.5 degrees) * 0.5 degrees
Using a calculator, sec^2(23.5 degrees) is approximately 1.1671.
delta_tan(0) = 1.1671 * 0.5 degrees
delta_tan(0) ≈ 0.5836 degrees
Therefore, the value of tan(0) is 0 = 23.5 degrees +/- 0.5836 degrees.
Step 2: Calculate d1 * tan(0)
To calculate d1 * tan(0) with uncertainty, we need to propagate the uncertainties using the formula:
delta(d1 * tan(0)) = (|tan(0)| * delta(d1)) + (|d1| * delta_tan(0))
To calculate delta(d1 * tan(0)), we substitute the given values and their uncertainties:
delta(d1 * tan(0)) = (|tan(23.5 degrees)| * 0.05 cm) + (|2.53 cm| * 0.5836 degrees)
In this case, we know that tan(23.5 degrees) is positive, so |tan(23.5 degrees)| = tan(23.5 degrees).
Using a calculator, tan(23.5 degrees) is approximately 0.4327.
delta(d1 * tan(0)) = (0.4327 * 0.05 cm) + (2.53 cm * 0.5836 degrees)
delta(d1 * tan(0)) ≈ 0.0216 cm + 1.4760 cm
delta(d1 * tan(0)) ≈ 1.4976 cm
Therefore, the value of d1 * tan(0) is 2.53 cm * 0.4327 +/- 1.4976 cm.
Please note that due to significant figures, the final result should be rounded to the appropriate number of decimal places based on the precision of the measurements.