Question

\(A\) force of \(500 \mathrm{nt}\) is measured with a possible error of \(1 \mathrm{nt}\). Its component in a direction \(60^{\circ}\) away from its line of action is required, where the angle is subject to an error of \(0,5^{\circ}\). What is (approximately) the largest possible error in the component?

Short answer

The largest possible error in the component is approximately 1 nt.

Step-by-step solution

- Understand the Input Data

The force is given as 500 newtons (nt) with a possible measurement error of 1 nt. The angle away from its line of action is 60 degrees with an error of 0.5 degrees.

- Express the Force Component

The component of the force in the direction 60 degrees away can be found using the formula: \[ F_x = F \times \text{cos}(\theta) \] where \( F = 500 \text{ nt} \) and \( \theta = 60° \).

- Compute the Nominal Component

Calculate the nominal force component: \[ F_x = 500 \times \text{cos}(60°) = 500 \times \frac{1}{2} = 250 \text{ nt} \].

- Consider the Error Impact on Force

Use the given possible error in force to calculate the maximum and minimum values of the force: \( F_{\text{max}} = 500 + 1 = 501 \text{ nt} \) and\( F_{\text{min}} = 500 - 1 = 499 \text{ nt} \).

- Recalculate Components with Force Error

Compute the components with these force values: \( F_x(\text{max}) = 501 \times \frac{1}{2} = 250.5 \text{ nt} \) and\( F_x(\text{min}) = 499 \times \frac{1}{2} = 249.5 \text{ nt} \).

- Consider the Error Impact on Angle

Calculate the angle errors: \( \theta_{\text{max}} = 60° + 0.5° = 60.5° \) and\( \theta_{\text{min}} = 60° - 0.5° = 59.5° \). Recompute the corresponding components.

- Recalculate Components with Angle Error

Compute the components with these angle values: \( F_x(\theta_{\text{max}}) = 500 \times \text{cos}(60.5°) \approx 500 \times 0.497 = 248.5 \text{ nt} \) and\( F_x(\theta_{\text{min}}) = 500 \times \text{cos}(59.5°) \approx 500 \times 0.502 = 251 \text{ nt} \).

- Determine Maximum Error in Component

The largest possible error in the component is coming from the maximum and minimum values, so: \( E_{\text{max}} = \text{max}(\left|250.5 - 250\right|, \left|249.5 - 250\right|, \left|248.5 - 250\right|, \left|251 - 250\right|) \) which simplifies to \( E_{\text{max}} = 1 \text{ nt} \).

Key concepts

Force Component Calculation

To find the component of a force along a particular direction, use the formula which involves the cosine trigonometric function. The formula is: \[ F_x = F \times \text{cos}(\theta) \], where \( F \) is the magnitude of the force and \( \theta \) is the angle between the force direction and the desired component direction.
In this specific exercise, the force is given as 500 newtons and the angle is 60 degrees. So, you plug these values into the formula: \[ F_x = 500 \times \text{cos}(60^{\text{circ}}) = 500 \times \frac{1}{2} = 250 \text{ nt} \].
This calculation gives you the nominal, or expected, value of the force component.

Angle Measurement Error

Measurement errors are quite common in physics and can significantly affect the results. In this exercise, the angle measurement has a possible error of \( \boldsymbol{0.5^{\text{circ}}} \). This means the actual angle could be either 60.5\text{^\text{circ}} or 59.5\text{^\text{circ}}.
To account for this, we recalculate the force component with these new angle values: \[ F_x(\theta_{\text{max}}) = 500 \times \text{cos}(60.5^{\text{circ}}) \text{ and } F_x(\theta_{\text{min}}) = 500 \times \text{cos}(59.5^{\text{circ}}) \].
Using a calculator or trigonometric table, we find: \[ 500 \times \text{cos}(60.5^{\text{circ}}) \text{ is approximately } 248.5 \text{ nt} \text{ and } 500 \times \text{cos}(59.5^{\text{circ}}) \text{ is approximately } 251 \text{ nt} \]. These two new values show us the potential range the force component might occupy due to the angle measurement error.

Maximum Possible Error

Lastly, to determine the maximum possible error in the force component, we look at both the errors due to the force magnitude and the angle measurement.
The force error gives us force values of \( 499 \text{ nt} \text{ and } 501 \text{ nt} \), and their components are recalculated as \( 249.5 \text{ nt} \) and \( 250.5 \text{nt} \), respectively.
Next, combining the angle errors, we already have the component values \( 248.5 \text{ nt} \text{ and } 251 \text{ nt} \).
The maximum error is the largest deviation from the nominal value of \( 250 \text{ nt} \). So, we calculate:
\[ E_{\text{max}} = \text{max}(|250.5 - 250|, |249.5 - 250|, |248.5 - 250|, |251 - 250|) \].
This results in \(\boldsymbol{1 \text{ nt}} \) as the maximum possible error in the force component.
Thus, given the uncertainties in both force and angle, the largest possible error is \(\boldsymbol{1 nt}\).