What is the algebraic expression for the uncertainty in the centripetal acceleration σac for uniform circular motion in terms of the uncertainties in the period and the radius of the motion? (Use the following as necessary: σT, σr, ac , T, and r.)
ac = 4 pi^2*r/T^2
ln(ac) = ln(4 pi^2) + ln r - 2 lnT
Now take the differential.
d(ac)/ac = lnr/r - 2 dT/T
For the standard deviation of the relative error, error, this leads to:
[σ(ac)/ac]^2 = [σr/r]^2 + [2*σT/T]^2
The relative error in ac, squared, equals the sum of the square of the relative error of r and the square of twice the relative error in T.
This assumes that errors in T and r are uncorrelated. Sometimes this is called the RSS (Root of the Sum of the Squares) rule.
To find the algebraic expression for the uncertainty in the centripetal acceleration (σac) for uniform circular motion in terms of the uncertainties in the period (σT) and the radius (σr) of the motion, you can use the method of partial derivatives and error propagation.
The formula for the centripetal acceleration (ac) in terms of the period (T) and the radius (r) is given by:
ac = (2πr) / T
To find the uncertainty in the centripetal acceleration, you need to differentiate this equation with respect to both variables, T and r.
Let's start by finding the partial derivative with respect to T:
∂ac/∂T = ∂(2πr / T) / ∂T
To solve this, we can use the quotient rule of differentiation:
∂(2πr / T) / ∂T = [(T * ∂(2πr) / ∂T) - (2πr * ∂T / ∂T)] / (T^2)
The second term in the numerator simplifies to:
∂T / ∂T = 1
Next, we need to find the partial derivative of (2πr) with respect to T:
∂(2πr) / ∂T = 0
Since the radius (r) is constant in this case, the derivative of r with respect to T will be zero.
Therefore, the partial derivative of ac with respect to T is:
∂ac/∂T = - (2πr) / (T^2)
Now, let's find the partial derivative with respect to r:
∂ac/∂r = ∂(2πr / T) / ∂r
To solve this, we can again use the quotient rule of differentiation:
∂(2πr / T) / ∂r = [(T * ∂(2πr) / ∂r) - (2πr * ∂T / ∂r)] / (T^2)
The second term in the numerator simplifies to:
∂T / ∂r = 0
Since the period (T) is constant in this case, the derivative of T with respect to r will be zero.
Next, we need to find the partial derivative of (2πr) with respect to r:
∂(2πr) / ∂r = 2π
Therefore, the partial derivative of ac with respect to r is:
∂ac/∂r = (2π) / T
Now, we can use error propagation to find the algebraic expression for σac in terms of σT and σr.
The general formula for error propagation is:
σf = sqrt((∂f/∂x)^2 * σx^2 + (∂f/∂y)^2 * σy^2 + ...)
Applying this formula for σac, we have:
σac = sqrt((∂ac/∂T)^2 * σT^2 + (∂ac/∂r)^2 * σr^2)
Plugging in the values we found earlier:
σac = sqrt((-(2πr) / T^2)^2 * σT^2 + ((2π) / T)^2 * σr^2)
Simplifying further:
σac = sqrt((4π^2r^2 / T^4) * σT^2 + (4π^2 / T^2) * σr^2)
Therefore, the algebraic expression for the uncertainty in the centripetal acceleration (σac) for uniform circular motion in terms of the uncertainties in the period (σT) and the radius (σr) of the motion is:
σac = sqrt((4π^2r^2 / T^4) * σT^2 + (4π^2 / T^2) * σr^2)