Torque is best defined as the tendency of a force to rotate an object around an axis, fulcrum, or pivot. Torque can be calculated using force and moment arm (the perpendicular distance from an axis to the line of action of a force) or by means of moment of inertia and angular acceleration.
Steps
Method 1 of 2: Use the Force and the Arm of the Moment
Step 1. Identify the forces exerted on the body and the corresponding moment arms
If the force is not perpendicular to the arm of the moment under consideration (i.e. it is mounted at an angle), it may be necessary to find the components using trigonometric functions such as sine or cosine.
- The component of the force you consider will depend on the equivalent of the perpendicular force.
- Imagine a horizontal bar and apply a force of 10N at an angle of 30 ° above the horizontal to rotate the body around its center.
- Since you have to use a force that is perpendicular to the moment arm, you need a vertical force to rotate the bar.
- Therefore, you have to consider the y component or use F = 10 sin30 ° N.
Step 2. Use the equation for the torque, τ = Fr where you simply replace the variables with the data you got or already have
- A simple example: imagine a 30 kg child sitting at the end of a swing. The length of the swing is 1.5m.
- Since the swing axis of rotation is in the center, you don't have to multiply by the length.
- You have to determine the force exerted by the child, using mass and acceleration.
- Since you have mass, you need to multiply it by the acceleration of gravity, g, which equals 9.81 m / s2.
- Now, you have all the data you need to use the torque equation:
Step 3. Use the sign conventions (positive or negative) to show the direction of the pair
When the force rotates the body clockwise, the torque is negative. When you turn it counterclockwise, the torque is positive.
- For multiple forces applied, you have to add up all the torques in the body.
- Since each force tends to produce rotations in different directions, the conventional use of the sign is important for keeping track of which forces act in which directions.
- For example, two forces F1 = 10, 0 N clockwise and F2 = 9, 0 N counterclockwise, are applied to the edge of a 0,050m diameter wheel.
- Since the given body is a circle, its fixed axis is the center. You have to halve the diameter to get the radius. The measurement of the radius will serve as the arm of the moment. So the radius is 0, 025 m.
- For clarity, we can solve for the individual torques generated by the forces.
- For force 1, the action is clockwise, so the torque produced is negative.
- For force 2, the action is counterclockwise, so the torque produced is positive.
- Now we can just add the pairs together to get the resulting pair.
Method 2 of 2: Use Moment of Inertia and Angular Acceleration
Step 1. Try to understand how the body's moment of inertia works to start solving the problem
The moment of inertia is the resistance of a body to rotational movement. It depends on the mass and also on how it is distributed.
- To understand this clearly, imagine two cylinders of the same diameter but of different masses.
- Imagine having to rotate the two cylinders with respect to their centers.
- Obviously, the cylinder with the higher mass will be more difficult to rotate than the other, as it is "heavier".
- Now imagine two cylinders with different diameters but the same mass. They will still appear with the same mass, but at the same time, having different diameters, the shapes or mass distributions of both cylinders will differ.
- The cylinder with a larger diameter will look like a flat, circular plate, while the smaller diameter cylinder will look like a tube of very compact consistency.
- The cylinder with a larger diameter will be more difficult to rotate, because you will need more force to account for the arm of the longest moment.
Step 2. Choose which equation to use to find the moment of inertia
There are several.
- First there is the simple equation with the sum of the mass and the moment arms of each particle.
- This equation is used for ideal points or particles. A material point is an object that has mass, but does not take up space.
- In other words, the only relevant feature of the object is its mass; it is not necessary to know its size, shape or structure.
- The concept of a material point is commonly used in physics to simplify calculations and use ideal and theoretical scenarios.
- Now, imagine objects like a hollow cylinder or a uniformly solid sphere. These objects have clear and precise shape, size and structure.
- Therefore, it is not possible to consider them as a material point.
- Thankfully, you can use the available equations that apply to some of these common objects.
Step 3. Find the moment of inertia
To start finding the torque, you need to calculate the moment of inertia. Use the following example problem:
- Two small "weights" of mass 5, 0 and 7, 0 kg are mounted at opposite ends of a 4.0 m long light bar (whose mass can be neglected). The axis of rotation is in the center of the rod. The rod is rotated starting from the state of rest with an angular velocity of 30.0 rad / s for 3, 00 s. Calculate the torque produced.
- Since the axis of rotation is in the center, the moment arm of both weights is equal to half the length of the rod, which is 2.0 m.
- Since the shape, size and structure of the "weights" were not specified, we can assume that they are ideal particles.
- The moment of inertia can be calculated as follows.
Step 4. Find the angular acceleration, α
The formula, α = at / r, can be used to calculate the angular acceleration.
- The first formula, α = at / r, can be used if the tangential acceleration and radius are known.
- Tangential acceleration is the acceleration tangent to the path of motion.
- Imagine an object along a curved path. Tangential acceleration is simply its linear acceleration at any point along the path.
- For the second formula, the simplest way to illustrate this concept is to relate it to kinematics: displacement, linear velocity, and linear acceleration.
- Displacement is the distance traveled by an object (SI unit: meter, m); linear velocity is the rate of change of the displacement over time (unit of measurement: m / s); linear acceleration is the rate of change of linear speed over time (unit of measurement: m / s2).
- Now, consider the counterparts in rotary motion: the angular displacement, θ, angle of rotation of a given point or line (SI unit: rad); the angular velocity, ω, variation of angular displacement over time (SI unit: rad / s); angular acceleration, α, change in angular velocity in the unit of time (SI unit: rad / s2).
- Going back to our example, you have been given the data for angular momentum and time. Since it started from a standstill, the initial angular velocity is 0. We can use the following equation for the calculation.
Step 5. Use the equation, τ = Iα, to find the torque
Simply replace the variables with the answers from the previous steps.
- You may notice that the unit "rad" does not fit into our units, because it is considered a dimensionless quantity, that is, without dimensions.
- This means that you can ignore it and continue with the calculation.
- For the sake of dimensional analysis, we can express the angular acceleration in the unit s-2.
Advice
- In the first method, if the body is a circle and the axis of rotation is the center, it is not necessary to find the components of the force (provided that the force is not inclined), since the force lies on the tangent of the circle immediately perpendicular to the arm of the moment.
- If you find it difficult to imagine how the rotation occurs, use the pen and try to recreate the problem. Be sure to copy the position of the axis of rotation and the direction of the applied force for a more adequate approximation.
