Calculate rotational torque using the torque equation τ = r·F·sin(θ). Determine torque from force, lever arm distance, and angle with step-by-step physics solutions for rotational mechanics.
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0° to 90° (90° = perpendicular)
0° to 90° (90° = perpendicular)
0° to 90° (90° = perpendicular)
Torque (τ)
0
N·m
Force (F)
—
Newtons (N)
Distance (r)
—
meters (m)
Mode
Torque
Calculation method used
📝 Step-by-Step Solution
Real-World Torque Examples
🔧 Wrench Tightening a Bolt
Problem: A mechanic applies a force of 40 N at the end of a 0.3 m wrench perpendicular to the handle. What torque is applied to the bolt?
Solution: Using τ = r·F·sin(θ) with θ = 90° (sin 90° = 1)
τ = 0.3 × 40 × 1 = 12 N·m
If the mechanic used a 0.6 m wrench with the same force, the torque would double to 24 N·m — a longer lever arm provides more torque.
⚖️ Seesaw Balancing
Problem: A child weighing 300 N sits 2 meters from the pivot of a seesaw. What torque does the child produce?
Solution: Using τ = r·F·sin(θ). The child's weight acts at 90° to the seesaw.
τ = 2 × 300 × sin(90°) = 2 × 300 × 1 = 600 N·m
For the seesaw to balance, an equal torque must be applied on the opposite side — this is why the distance from the pivot matters so much.
🚗 Engine Torque
Problem: A car engine produces 400 N·m of torque at the crankshaft. The connecting rod has a lever arm of 0.12 m and the force is applied at 75°. What force does the piston exert?
F = 400 / (0.12 × sin(75°)) = 400 / (0.12 × 0.9659) = 400 / 0.1159 = 3,450 N
This shows how the crankshaft converts linear piston force into rotational torque — a fundamental principle in all internal combustion engines.
🔩 Bolt Removal with an Angle
Problem: You need 50 N·m of torque to loosen a bolt. You can apply 100 N of force at a 60° angle on a wrench. What lever arm distance is needed?
Solution: Using r = τ / (F·sin(θ))
r = 50 / (100 × sin(60°)) = 50 / (100 × 0.8660) = 50 / 86.60 = 0.577 m
At a 90° angle, only 0.5 m would be needed. The smaller angle requires a longer wrench to achieve the same torque — always apply force perpendicular to the lever arm for maximum efficiency.
Torque Formula & Guide
τ = r × F × sin(θ)
Torque equation — τ (torque), r (lever arm), F (force), θ (angle)
Where τ is torque in N·m, r is the distance from the pivot point (lever arm) in meters, F is the applied force in Newtons, and θ is the angle between the force vector and the lever arm.
F = τ / (r × sin(θ))
Force needed to produce a given torque
Rearrange the torque equation to solve for force when you know the required torque, lever arm distance, and application angle.
r = τ / (F × sin(θ))
Lever arm distance needed for a given torque and force
Solve for the lever arm distance when you know the torque, applied force, and angle. Useful for determining tool length requirements.
Key Concepts
🔩 What is Torque?
Torque (also called moment of force) is the rotational equivalent of linear force. It measures the tendency of a force to cause an object to rotate about an axis or pivot point. The SI unit is the Newton-meter (N·m).
📐 The Angle Matters
The sin(θ) factor in the torque equation means that maximum torque occurs when the force is applied perpendicular (90°) to the lever arm. As the angle decreases, less torque is produced for the same force. At 0°, the force is parallel to the lever arm and produces zero torque.
🔧 Lever Arm Principle
Increasing the lever arm distance (r) proportionally increases the torque. This is why longer wrenches make it easier to loosen tight bolts — they provide more torque for the same applied force. This principle is also used in crowbars, wheelbarrows, and other simple machines.
⚡ Torque vs. Power
Torque and power are related but distinct concepts. Torque is a rotational force, while power is the rate at which work is done. In engines, power = torque × angular velocity (P = τ × ω). High torque is important for acceleration, while high power determines top speed.
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Torque Mode (τ = r·F·sinθ)
Calculate rotational torque from force, lever arm distance, and application angle. Supports N·m, kN·m, lb·ft, and lb·in output units.
⚡
Force Mode (F = τ/(r·sinθ))
Determine the force required to produce a specific torque given the lever arm distance and angle. Supports N, kN, and lbf force units.
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Distance Mode (r = τ/(F·sinθ))
Find the necessary lever arm distance to achieve a target torque with a known force and angle. Supports m, cm, ft, and in distance units.
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Step-by-Step Solutions
Every calculation comes with a detailed step-by-step breakdown showing the formula, substitution with SI unit conversions, intermediate values, and the final result.
⚠️ Important Note: This calculator assumes a simplified torque model. Real-world rotational mechanics may involve additional factors such as friction, distributed loads, multiple forces, non-rigid bodies, and dynamic effects. For critical engineering applications, always verify results with physical testing and consult relevant mechanical engineering standards.
Frequently Asked Questions
What is torque in physics?
Torque (also called the moment of force) is the rotational equivalent of linear force. It measures how much a force causes an object to rotate about an axis. Mathematically, torque is defined as the cross product of the lever arm vector and the force vector: τ = r × F, with magnitude τ = r·F·sin(θ). The SI unit is the Newton-meter (N·m). Torque is a vector quantity — its direction is perpendicular to both the lever arm and force vectors (right-hand rule).
How do I calculate torque from force and distance?
To calculate torque, use the formula τ = r × F × sin(θ). Measure the distance r from the pivot point to where the force is applied. Measure the force F in Newtons. Determine the angle θ between the force vector and the lever arm. For maximum torque, apply the force perpendicular to the lever arm (θ = 90°, sin 90° = 1). For example, a 30 N force applied 0.4 m from the pivot at 90° produces τ = 0.4 × 30 × 1 = 12 N·m.
What is the difference between torque and force?
Force is a push or pull that causes linear acceleration (F = ma). Torque is a twisting force that causes rotational acceleration (τ = I·α, where I is moment of inertia and α is angular acceleration). Force is measured in Newtons (N) and produces linear motion. Torque is measured in Newton-meters (N·m) and produces rotational motion. A force can exist without producing torque if it acts directly through the pivot point (r = 0 or θ = 0°).
Why does the angle matter in torque calculations?
The angle θ between the force vector and the lever arm determines what fraction of the force contributes to rotation. Only the component of force perpendicular to the lever arm produces torque (F_perp = F·sinθ). When θ = 90°, all of the force contributes to torque (maximum). When θ = 0°, the force is parallel to the lever arm and produces zero torque. For angles between 0° and 90°, only a portion of the force creates torque. This is why you always want to push or pull perpendicular to a wrench handle for maximum efficiency.
What units are used for torque?
The SI unit of torque is the Newton-meter (N·m). Other common units include kilonewton-meters (kN·m) for large torques (like engine torque), pound-feet (lb·ft) commonly used in the US for automotive and mechanical applications, and pound-inches (lb·in) for smaller torque values. In some contexts, torque is also expressed in dyne-centimeters (dyn·cm) in the CGS system. Note that while N·m has the same units as energy (Joules), torque is fundamentally different — torque is a vector, while energy is a scalar.
How is torque related to rotational motion?
Torque is to rotational motion what force is to linear motion. The rotational analog of Newton's Second Law is τ = I × α, where I is the moment of inertia (rotational mass) and α is angular acceleration. Just as force causes linear acceleration, torque causes angular acceleration. Torque is also related to work: W = τ × θ (torque multiplied by angular displacement in radians), and to power: P = τ × ω (torque multiplied by angular velocity). This relationship is crucial in understanding engines, motors, and all rotating machinery.