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Torque Formula Calculator

\[ \tau = r F \sin(\theta) \]

1. What is the Torque Formula Calculator?

Definition: This calculator computes the torque (\(\tau\)) exerted by a force applied at a distance from a pivot point, using the formula \(\tau = r F \sin(\theta)\), where \(r\) is the distance (lever arm), \(F\) is the force, and \(\theta\) is the angle between the force and the lever arm.

Purpose: It is used in mechanics to determine the rotational force on an object, applicable in engineering, physics, and mechanical design.

2. How Does the Calculator Work?

The calculator uses the torque formula:

Formula: \[ \tau = r F \sin(\theta) \] where:

  • \(\tau\): Torque (N·m, kN·m, lb·ft)
  • \(r\): Distance (m, cm, ft, in)
  • \(F\): Force (N, kN, lbf)
  • \(\theta\): Angle (rad, deg)

Unit Conversions:

  • Distance:
    • 1 m = 1 m
    • 1 cm = 0.01 m
    • 1 ft = 0.3048 m
    • 1 in = 0.0254 m
  • Force:
    • 1 N = 1 N
    • 1 kN = 1000 N
    • 1 lbf = 4.4482216152605 N
  • Angle:
    • 1 rad = 1 rad
    • 1 deg = \( \frac{\pi}{180} \) rad \(\approx 0.01745329252 \, \text{rad}\)
  • Torque (Output):
    • 1 N·m = 1 N·m
    • 1 kN·m = 1000 N·m
    • 1 lb·ft = 1.3558179483314 N·m
The torque is calculated in N·m and can be converted to the selected output unit (N·m, kN·m, lb·ft). Results greater than 10,000 or less than 0.001 are displayed in scientific notation; otherwise, they are shown with 4 decimal places.

Steps:

  • Enter the distance (\(r\)), force (\(F\)), and angle (\(\theta\)) with their units (default: \(r = 2 \, \text{m}\), \(F = 10 \, \text{N}\), \(\theta = 90^\circ\)).
  • Convert inputs to SI units (m, N, rad).
  • Validate that distance is greater than 0 and angle is within the appropriate range (0 to 360° or 0 to \(2\pi\) rad).
  • Calculate the torque in N·m using the formula.
  • Convert the torque to the selected output unit.
  • Display the result, using scientific notation if the value is greater than 10,000 or less than 0.001, otherwise rounded to 4 decimal places.

3. Importance of Torque Calculation

Calculating torque is crucial for:

  • Mechanics: Understanding rotational motion in systems like levers, gears, and engines.
  • Engineering: Designing mechanical systems, such as motors and machinery, where torque determines rotational force.
  • Education: Teaching the principles of rotational dynamics and the role of force, distance, and angle in torque.

4. Using the Calculator

Examples:

  • Example 1: Calculate the torque for \(r = 2 \, \text{m}\), \(F = 10 \, \text{N}\), \(\theta = 90^\circ\), output in N·m:
    • Enter \(r = 2 \, \text{m}\), \(F = 10 \, \text{N}\), \(\theta = 90^\circ\).
    • Convert: \(\theta = 90 \times \frac{\pi}{180} = \frac{\pi}{2} \, \text{rad}\).
    • Angle factor: \(\sin(\theta) = \sin(\frac{\pi}{2}) = 1\).
    • Torque: \(\tau = 2 \times 10 \times 1 = 20 \, \text{N·m}\).
    • Output unit: N·m (no conversion needed).
    • Result: \( \text{Torque} = 20.0000 \, \text{N·m} \).
  • Example 2: Calculate the torque for \(r = 6.56168 \, \text{ft}\), \(F = 2.24809 \, \text{lbf}\), \(\theta = 1.5708 \, \text{rad}\), output in lb·ft:
    • Enter \(r = 6.56168 \, \text{ft}\), \(F = 2.24809 \, \text{lbf}\), \(\theta = 1.5708 \, \text{rad}\).
    • Convert: \(r = 6.56168 \times 0.3048 = 2 \, \text{m}\), \(F = 2.24809 \times 4.4482216152605 = 10 \, \text{N}\).
    • Angle factor: \(\sin(\theta) = \sin(1.5708) \approx 1\).
    • Torque in N·m: \(\tau = 2 \times 10 \times 1 = 20 \, \text{N·m}\).
    • Convert to output unit (lb·ft): \(20 \times \frac{1}{1.3558179483314} \approx 14.7513 \, \text{lb·ft}\).
    • Result: \( \text{Torque} = 14.7513 \, \text{lb·ft} \).

5. Frequently Asked Questions (FAQ)

Q: What is torque?
A: Torque is a measure of rotational force, representing the tendency of a force to rotate an object about a pivot point, calculated as the product of the force, the distance from the pivot, and the sine of the angle between them.

Q: Why must distance be greater than zero?
A: Zero or negative distance is physically meaningless in this context, as it represents the lever arm’s length from the pivot point, which must be positive for torque to be defined.

Q: What does the angle \(\theta\) represent?
A: The angle \(\theta\) is the angle between the force vector and the lever arm (the line from the pivot to the point of force application). Maximum torque occurs when \(\theta = 90^\circ\) (\(\sin(\theta) = 1\)), and zero torque occurs when \(\theta = 0^\circ\) or \(180^\circ\) (\(\sin(\theta) = 0\)).

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