Torque

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: Torque is the rotational force applied to an object. The magnitude of torque depends on the rotational mass of the object and its distance from its axis, or point of rotation. Torque is relevant to many industrial and household machines.

Principal Terms

• angular momentum: the rotational momentum of an object around an axis, defined as the product of its moment and its angular velocity.
• axis: the center around which an object rotates.
• cross product: an operation, broadly analogous to multiplication, performed on two vectors in a three-dimensional space that results in a third vector that is perpendicular to both; if both vectors have the same direction or if one of them has a value of zero, the cross product will be zero.
• fulcrum: the supporting point around which a lever pivots.
• mechanical advantage: a measurement of the increase in force achieved by applying a mechanical tool or device to an existing system.
• moment: a combination of a physical quantity and a distance with respect to a fixed axis; the physical quantities of an object as measured at some distance from that axis.
• vector: a quantity that has direction as well as magnitude.

Linear and Angular Momentum

All moving objects have momentum. An object’s momentum is simply a measure of how much motion it has. The greater the momentum in a given direction, the more the object tends to continue moving in that direction. If an object is moving in a straight line, its linear momentum is equal to the product of its mass and its velocity.

Momentum and velocity are both vector quantities. Therefore, an object moving linearly has either a positive or negative velocity, depending on which way it is moving relative to its starting point. For example, if a car drives north at a velocity of 20 meters per second, north can be considered the positive direction. Then, if the car turns around and drives back to its starting point at the same speed, it will have a velocity of −20 meters per second.

However, most objects do not move only in straight lines. Many rotate around some kind of a fixed point, or axis. An object rotating around an axis has angular momentum. Angular momentum, like linear momentum, is the product of an object’s mass and its velocity, but it is measured in different units. An object’s angular mass, or moment of inertia, is how much it resists changing its angular velocity around its axis. Its angular velocity is the rate at which its angle around the axis changes. Thus, the angular momentum of an object is equal to the product of its moment of inertia and its angular velocity.

According to Isaac Newton’s second law of motion, momentum stays constant if no outside force acts on a given system, as the rate of change in momentum is zero. However, once an outside force interacts with a system, momentum can increase or decrease, depending on the direction and magnitude of the applied force. When the system involves rotation, that outside force is called torque.

Torque, Force, and Displacement Vectors

Torque is the degree to which a force causes an object to rotate around an axis. It can also be thought of as an object’s change in angular momentum, or a "twist" applied to a moving object. For example, a force applied to a door will cause the door to rotate around its hinges rather than moving in a straight line. The hinges serve as the door’s axis. The rotational part of the applied force is the torque.

Mathematically, torque is calculated as

T = F × r

where T is torque, F is the rotational force, and r is the displacement vector, which measures the distance between the axis and the point of force application. (Vector quantities are represented by bolded variables to distinguish them from scalar quantities, which have magnitude but no direction.) The unit of torque is the newton-meter (N∙m), a compound unit of force and distance. By convention, torque is considered negative if the direction of rotation is clockwise and positive if it is counterclockwise.

Torque is important to many household tools and machines. Most of these are based on the six classical simple machines that use mechanical advantage to multiply force. Levers offer the best example of torque. A lever consists of a flat surface attached to a fulcrum. When no net torque is applied, the surface balances on the fulcrum. However, if more force is applied to one side than the other, it will produce a torque that causes the surface to rotate. The position of the fulcrum is key. As the distance from the fulcrum to the point of force application increases, the amount of force required to rotate the plane around the fulcrum decreases. This is because the lever applies a rotational torque that multiplies the force. For example, pushing on a door (a type of lever) very close to its hinges takes much more force than pushing farther away from the hinges. The longer the radius, the greater the torque applied. Similarly, another simple machine, the screw, can translate the rotational force of torque into an amplified linear force.

Direction and Magnitude: Cross Products and Angles

When multiplying vector quantities, direction and magnitude must be calculated separately. Because torque is the cross product of two vectors (F and r) in a three-dimensional space, its direction is perpendicular to both. For example, if the string of a yo-yo is pinned to a tabletop and a force is pushing the yo-yo to rotate around the pin, the yo-yo’s displacement vector points along the string and its force points along its spinning edge. The torque vector is thus perpendicular to both of them, pointing straight up in the air.

When calculating the magnitude of torque, the angle at which it is applied must be taken into account. By definition, the vector F in the torque equation represents only the part of the force perpendicular to the displacement vector r. The full equation for the magnitude of torque, not taking direction into account, is

T = Frsinθ

where θ is the angle between the vectors F and r. For example, if seventeen newtons of force are applied to a door at a point 1.5 meters from its hinges, and the angle of force application is sixty degrees from the flat of the door, then the magnitude of the torque would be calculated as follows:

If the force is applied perpendicular to door, however, so that θ equals ninety degrees, then sinθ is equal to 1 and can be disregarded.

Calculating Torque

Before calculating the torque of an object, the axis of rotation must be determined. For example, if the object is a lever, such as a seesaw, then the axis of rotation is the fulcrum. Next, the directions and magnitudes of the forces acting on the object must be identified. If two children are sitting on opposite sides of the seesaw, each child exerts a downward force on the lever that is a combination of the child’s mass and the force of gravity. If the system in which the object exists is in equilibrium—for example, if the seesaw is perfectly balanced—then the net torque is be zero. If the system is not in equilibrium and an external force is acting on it to produce rotation, the torque can be calculated by using the equations above.

Sample Problem

A bolt is tightened with a torque of 17 newton-meters (N·m). Two different wrenches are available to turn the bolt. One is 20 centimeters (cm) long, and the other 30 cm long. If force is applied to both wrenches at an angle of 30 degrees, what is the difference in the amount of force needed to turn the bolt with each wrench?

Answer:

To find the difference in force between the two wrenches, the force needed for each wrench to turn the bolt must first be calculated. Because the turning force is applied at an angle of 30 degrees, the perpendicular component of the force is equal to the sine of 30 degrees:

sin30° = sin(π/6) = 1/2

Substitute all the known quantities for the 20-centimeter wrench into the equation for magnitude of torque:

T = Frsinθ

17 N·m = F(20 cm)(1/2)

Convert the centimeters to meters and then solve for F:

17 N·m = F(0.2 m)(1/2)

17 N·m = F(0.1 m)

(17 N·m) / (0.1 m) = F

170 N = F

The 20-centimeter wrench requires an application of 170 N of force. Repeat the calculations for the 30-centimeter wrench:

17 N·m = F(0.3 m)(1/2)

17 N·m = F(0.15 m)

(17 N·m) / (0.15 m) = F

113.33 N = F

Calculate the difference:

170 N – 113.33 N = 56.67 N

The 30-centimeter wrench requires 56.67 N less force to turn the bolt.

Torque in Daily Life

Torque is critical to many aspects of engineering and mechanical design, from static structures to complex machines. A car’s engine, for example, produces torque from the combustion of gas. That torque turns the crankshafts, which then turn the wheels. The amount of torque ascribed to an engine reflects how quickly it can accelerate the car. That torque must overcome friction and air drag, among other forces, in order to move the car forward.

Torque can also be used to calculate an engine’s power (P), or the amount of work it can perform over time, if the angular velocity (ω) of its output shaft is also known:

P = Tω

This is especially useful for finding the power output of an electric motor.

For static structures, connecting elements (e.g., metal bolts that hold sheets of steel together) must be able to withstand the torque of potential outside forces, such as the weight of cars on the bridge or the strength of a storm’s winds against the building.

Bibliography

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Khan, Salman. "Moments, Torque, and Angular Momentum." Khan Academy. Khan Acad., 23 May 2008. Web. 7 Aug. 2015.

Nave, Carl R. "Torque." HyperPhysics. Georgia State U, 2012. Web. 7 Aug. 2015.

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Roldán Cuenya, Beatriz. "Chapter 11: Torque and Angular Momentum." Physics for Scientists and Engineers I.U of Central Florida, 2013. Web. 7 Aug. 2015.

"Torque and Levers." Siyavula Textbooks: Grade 11 Physical Science. Rice U, 29 July 2011. Web. 7 Aug. 2015.