Calculating Torque With Examples

Calculating Torque With Examples When studying how objects rotate, it quickly becomes necessary to figure out how a given force results in a change in the rotational motion. The tendency of a force to cause or change rotational motion is called torque, and its one of the most important concepts to understand in resolving rotational motion situations. The Meaning of Torque Torque (also called moment - mostly by engineers) is calculated by multiplying force and distance. The SI units of torque are newton-meters, or N*m (even though these units are the same as Joules, torque isnt work or energy, so should just be newton-meters). In calculations, torque is represented by the Greek letter tau: Ï„. Torque is a vector quantity, meaning it has both a direction and a magnitude. This is honestly one of the trickiest parts of working with torque because it is calculated using a vector product, which means you have to apply the right-hand rule. In this case, take your right hand and curl the fingers of your hand in the direction of rotation caused by the force. The thumb of your right hand now points in the direction of the torque vector. (This can occasionally feel slightly silly, as youre holding your hand up and pantomiming in order to figure out the result of a mathematical equation, but its the best way to visualize the direction of the vector.) The vector formula that yields the torque vector Ï„ is: Ï„ r Ãâ€" F The vector r is the position vector with respect to an origin on the axis of rotation (This axis is the Ï„ on the graphic). This is a vector with a magnitude of the distance from where the force is applied to the axis of rotation. It points from the axis of rotation toward the point where the force is applied. The magnitude of the vector is calculated based upon ÃŽ ¸, which is the angle difference between r and F, using the formula: Ï„ rFsin(ÃŽ ¸) Special Cases of Torque A couple of key points about the above equation, with some benchmark values of ÃŽ ¸: ÃŽ ¸ 0 ° (or 0 radians) - The force vector is pointing out in the same direction as r. As you might guess, this is a situation where the force will not cause any rotation around the axis ... and the mathematics bears this out. Since sin(0) 0, this situation results in Ï„ 0.ÃŽ ¸ 180 ° (or Ï€ radians) - This is a situation where the force vector points directly into r. Again, shoving toward the axis of rotation isnt going to cause any rotation either and, once again, the mathematics supports this intuition. Since sin(180 °) 0, the value of the torque is once again Ï„ 0.ÃŽ ¸ 90 ° (or Ï€/2 radians) - Here, the force vector is perpendicular to the position vector. This seems like the most effective way that you could push on the object to get an increase in rotation, but does the mathematics support this? Well, sin(90 °) 1, which is the maximum value that the sine function can reach, yielding a result of Ï„ rF. In other words, a force applied at any other angle would provide less torque than when it is applied at 90 degrees. The same argument as above applies to cases of ÃŽ ¸ -90 ° (or -Ï€/2 radians), but with a value of sin(-90 °) -1 resulting in the maximum torque in the opposite direction. Torque Example Lets consider an example where youre applying a vertical force downward, such as when trying to loosen the lug nuts on a flat tire by stepping on the lug wrench. In this situation, the ideal situation is to have the lug wrench perfectly horizontal, so that you can step on the end of it and get the maximum torque. Unfortunately, that doesnt work. Instead, the lug wrench fits onto the lug nuts so that it is at a 15% incline to the horizontal. The lug wrench is 0.60 m long until the end, where you apply your full weight of 900 N. What is the magnitude of the torque? What about direction?: Applying the lefty-loosey, righty-tighty rule, you will want to have the lug nut rotating to the left - counter-clockwise - in order to loosen it. Using your right hand and curling your fingers in the counter-clockwise direction, the thumb sticks out. So the direction of the torque is away from the tires ... which is also direction you want the lug nuts to ultimately go. To begin calculating the value of the torque, you have to realize that theres a slightly misleading point in the above set-up. (This is a common problem in these situations.) Note that the 15% mentioned above is the incline from the horizontal, but thats not the angle ÃŽ ¸. The angle between r and F has to be calculated. Theres a 15 ° incline from the horizontal plus a 90 ° distance from the horizontal to the downward force vector, resulting in a total of 105 ° as the value of ÃŽ ¸. Thats the only variable that requires set-up, so with that in place we just assign the other variable values: ÃŽ ¸ 105 °r 0.60 mF 900 N Ï„ rF sin(ÃŽ ¸) (0.60 m)(900 N)sin(105 °) 540 Ãâ€" 0.097 Nm 520 Nm Note that the above answer involved maintaining only two significant figures, so it is rounded. Torque and Angular Acceleration The above equations are particularly helpful when there is a single known force acting on an object, but there are many situations where a rotation can be caused by a force that cannot easily be measured (or perhaps many such forces). Here, the torque often isnt calculated directly, but can instead be calculated in reference to the total angular acceleration, ÃŽ ±, that the object undergoes. This relationship is given by the following equation: ÃŽ £Ãâ€ž - The net sum of all torque acting on the objectI - the moment of inertia, which represents the objects resistance to a change in angular velocityÃŽ ± - angular acceleration

How to Identify the Common Black Walnut Tree

How to Identify the Common Black Walnut Tree Black walnut trees (Juglan  nigra) are found throughout much of the central-eastern part of the U.S., except in the far northern and far southern part of this range, but familiar elsewhere from the East Coast into the central plains. They are part of the general plant family Juglandaceae, which includes all the walnuts as well as hickory trees. The Latin name, Juglans, derives from Jovis glans, Jupiters acornfiguratively, a nut fit for a god. There are 21 species in the genus that range across the north temperate Old World from southeast Europe east to Japan, and more widely in the New World from southeast Canada west to California and south to Argentina. There are five native walnut species in North America: black walnut, butternut, Arizona walnut and two species in California. The two most commonly found walnuts found in native locations are the black walnut and butternut.   In its natural setting, the black walnut favors riparian zonesthe transition areas between rivers, creeks and denser woods. It does best in sunny areas, as it is classified as shade intolerant.   The black  walnut  is known as an  allelopathic tree: it releases chemicals in the ground that may poison other plants. A black walnut can sometimes be identified by the dead or yellowing plants in its vicinity.   It often appears as a kind of weed tree along roadsides and in open areas, due to the fact that squirrels and other animals harvest and spread the nuts. It is often found in the same environment as silver maples, basswoods, white ash, yellow-poplar, elm and hackberry trees.   Description Walnuts are specifically deciduous trees, 30 to 130 feet tall with pinnate leaves containing five to 25 leaflets. The actual leaf is attached to twigs in a  mostly alternate arrangement and the leaf structure is  odd-pinnately  compound- meaning that the leaves consist of an odd number of individual leaflets that attach to a central stem. These leaflets are serrate or  toothed.  The shoots and twigs have a chambered pith, a characteristic that can quickly confirm the trees identification when a twig is cut open. The fruit of a walnut is a rounded, hard-shelled nut. Butternuts are similar, but  this type of native walnut has oblong  ridged  fruits that form in clusters. The leaf scars on butternut have a hairy top fringe, while walnuts do not. Identification When Dormant During dormancy, the black walnut can be identified by examining the bark; the leaf scars are seen when leaves are pulled away from branches, and by looking at the nuts that have fallen around the tree. In a black walnut, the bark is furrowed and dark in color (it is lighter in butternut). The leaf scars along twigs look  like an  upside-down shamrock with five or seven bundle scars. Beneath the tree, you usually find whole walnuts or their husks. The black walnut has a  globose nut (meaning it is roughly globular or round), while the nuts on the butternut tree are more egg-shaped and smaller.