Torque Calculator
Calculate Torque
Enter the force, lever arm length, and optionally the angle to calculate torque.
The magnitude of the force applied.
The distance from the pivot point to the point where force is applied.
What is Torque?
Torque, also known as moment of force, is a measure of the rotational force or twisting force that causes an object to rotate about an axis, fulcrum, or pivot. Just as a linear force is a push or a pull, torque can be thought of as a twist to an object. It’s a vector quantity, meaning it has both magnitude and direction.
The magnitude of torque depends on three factors: the magnitude of the force applied, the length of the lever arm (the distance from the axis of rotation to the point where the force is applied), and the angle between the force vector and the lever arm vector. The direction of the torque is perpendicular to both the force and the lever arm, determined by the right-hand rule.
Anyone working with rotating objects or machinery, such as engineers, mechanics, physicists, and even DIY enthusiasts, should understand and use the concept of torque. It’s crucial in designing engines, tightening bolts, using wrenches, and understanding the mechanics of rotating systems. Our torque calculator helps you easily determine this value.
A common misconception is that torque is the same as work or energy. While both torque and work involve force and distance, torque is a rotational force measured in Newton-meters (N·m) or pound-feet (lb·ft), and work is energy transferred, measured in Joules (J) or foot-pounds (ft·lb). One Newton-meter of torque is different from one Joule of work.
Torque Formula and Mathematical Explanation
The formula to calculate torque (τ) is:
τ = r × F = |r| |F| sin(θ)
Where:
- τ (tau) is the torque vector, and |τ| is its magnitude.
- r is the lever arm vector (or position vector) from the axis of rotation to the point where the force is applied, and |r| is its length.
- F is the force vector, and |F| is its magnitude.
- θ (theta) is the angle between the lever arm vector r and the force vector F.
- × denotes the cross product of the vectors.
The magnitude of the torque is given by:
|τ| = |r| |F| sin(θ)
If the force is applied perpendicular to the lever arm (θ = 90° or π/2 radians), then sin(θ) = 1, and the formula simplifies to:
|τ| = |r| |F|
Our torque calculator uses these formulas based on your inputs.
Variables Table
| Variable | Meaning | SI Unit | Imperial Unit | Typical Range |
|---|---|---|---|---|
| τ | Torque | Newton-meter (N·m) | Pound-foot (lb·ft) | 0 to thousands |
| F | Force | Newton (N) | Pound-force (lbf) | 0 to thousands |
| r | Lever Arm Length | meter (m) | foot (ft) | 0.01 to hundreds |
| θ | Angle | degrees (°) / radians (rad) | degrees (°) / radians (rad) | 0° to 180° (0 to π rad) |
Practical Examples (Real-World Use Cases)
Example 1: Tightening a Bolt
Imagine you are using a wrench to tighten a bolt. The wrench is 0.3 meters long, and you apply a force of 100 Newtons perpendicularly to the end of the wrench.
- Force (F) = 100 N
- Lever Arm Length (r) = 0.3 m
- Angle (θ) = 90° (perpendicular)
Using the formula |τ| = |r| |F| sin(90°) = 0.3 m * 100 N * 1 = 30 N·m. You are applying 30 N·m of torque to the bolt. You can verify this with our torque calculator.
Example 2: Pushing a Door
You push a door open. The doorknob is 0.8 meters from the hinges (pivot), and you push with a force of 20 Newtons at an angle of 60 degrees to the plane of the door (so 60 degrees to the lever arm from the hinge to the knob).
- Force (F) = 20 N
- Lever Arm Length (r) = 0.8 m
- Angle (θ) = 60°
|τ| = |r| |F| sin(60°) = 0.8 m * 20 N * sin(60°) ≈ 0.8 * 20 * 0.866 ≈ 13.86 N·m. The effective torque is about 13.86 N·m.
How to Use This Torque Calculator
- Select Units: Choose between Metric (Newtons, meters) and Imperial (pounds-force, feet) units for force and length using the dropdown.
- Enter Force: Input the magnitude of the force you are applying in the “Force” field.
- Enter Lever Arm Length: Input the length of the lever arm (distance from pivot to force application point).
- Choose Force Application: Select “Perpendicular (90°)” if the force is applied at a right angle to the lever arm. Select “Specify Angle” if you want to enter a custom angle.
- Enter Angle (if applicable): If you selected “Specify Angle”, enter the angle in degrees (between 0 and 180) between the force and lever arm.
- View Results: The calculator automatically updates the torque in N·m and lb·ft, along with intermediate values, as you type.
- Analyze Chart: The chart shows how torque varies with the angle for your given force and lever arm.
- Reset: Click “Reset” to clear inputs and go back to default values.
- Copy: Click “Copy Results” to copy the main results and inputs to your clipboard.
The results from the torque calculator will show the primary torque value and also convert it between N·m and lb·ft.
Key Factors That Affect Torque Results
- Magnitude of Force (F): The greater the force applied, the greater the torque, assuming the lever arm and angle remain constant. Double the force, and you double the torque.
- Length of the Lever Arm (r): The longer the lever arm, the greater the torque produced by the same force. This is why it’s easier to loosen a tight bolt with a longer wrench.
- Angle (θ) between Force and Lever Arm: Torque is maximized when the force is applied perpendicular (90°) to the lever arm (sin(90°) = 1). As the angle deviates from 90°, sin(θ) decreases, reducing the torque. At 0° or 180°, the torque is zero because the force is either pushing or pulling directly along the lever arm, causing no rotation.
- Point of Force Application: The lever arm length is measured from the axis of rotation to the point where the force is applied. Changing this point changes ‘r’.
- Direction of Force: While our calculator focuses on magnitude, the direction of the force determines the direction of the torque (clockwise or counter-clockwise), which is important in many applications.
- Units Used: Ensuring consistent units (e.g., Newtons and meters, or pounds-force and feet) is crucial for accurate calculations. Our torque calculator handles conversions for convenience.
Frequently Asked Questions (FAQ)
A: Torque is a rotational force (a twist), measured in N·m or lb·ft. Work is energy transferred by a force acting over a distance, measured in Joules (J) or ft·lb. Mathematically, work is the dot product of force and displacement, while torque is the cross product of lever arm and force.
A: The standard SI unit for torque is the Newton-meter (N·m). In the imperial system, it’s often pound-foot (lb·ft) or foot-pound (ft·lb), though ft·lb is also used for work, leading to potential confusion.
A: Torque is fundamental in understanding and designing anything that rotates, from engines and gears to simple tools like wrenches and screwdrivers. It determines the rotational acceleration an object will experience.
A: To maximize torque for a given force, apply the force as far as possible from the pivot point (maximize ‘r’) and at an angle as close to 90 degrees as possible to the lever arm.
A: Yes, the sign of the torque usually indicates its direction (e.g., clockwise or counter-clockwise). Our torque calculator provides the magnitude.
A: This calculator assumes a constant force. If the force varies, more advanced methods (like integration) are needed to find the total effect.
A: The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force. In our formula |τ| = |r| |F| sin(θ), |r|sin(θ) represents this perpendicular distance when |r| is the distance to the point of force application.
A: No, this calculator focuses on the torque produced by an applied external force. The weight of the lever arm itself could produce its own torque if its center of mass is not at the pivot.