chapter9


 * Chapter 9 Rotationtoc**

Link to following document: media type="custom" key="9460136"

Moment of Inertia
Link to following document: media type="custom" key="9464628"

Work and Power
media type="custom" key="9523818"

Torque
Torque is the rotational equivalent of force. It is defined as **τ =** **r** x **F**. As with all cross products, torque is a **vector** quantity. This is the cross-product of radius (distance) and force and can therefore also be represented as **rFsinθ**. Given the properties a cross product, a third form of this equation is **rF** **┴**. This means that any force which pulls perpendicular to the radius creates a torque equal to simply rF.
 * Note: these torque equations can also be defined with the variable //d// (for distance) instead of //r//, making the generic formula //Fd//. [[image:Torque_at_angle.png width="160" height="192" align="right"]]

Sign Convention: **Counterclockwise is positive. Clockwise is negative.** This means that forces that pull at an angle to rotate an object in the counterclockwise direction are recorded as positive toques. Conversely, forces pulling at an angle to rotate the object in a clockwise direction are recorded as negative torques.

Torque is **measured in Newton meters** (denoted **Nm**). While this is technically the same as the unit of joules (since work is also force times distance), different units are used because the values of the two measurements are not interchangeable.

In the diagram to the right, with a bar of length //r// rotating around the grey circle, the force //F// would rotate it in the clockwise direction and therefore creates a negative torque. The torque is equal to **rF** **┴**. In this example, the perpendicular force is actually Fcosθ (not Fsinθ as the cross product would suggest). This is because the angle //θ// given is NOT the angle between the force and the bar being rotated. So, the final torque equation for this system can be written as //rF//cos//θ//.

Like many other linear equations, Newtons Second Law (F = ma) is also translatable to angular terms. Given that torque is the rotational analogue of force**, τ = Iα **. This makes sense conceptually because, as discussed previously on this page, moment of inertia the rotational equivalent of mass and is the resisting factor in a change of momentum. Therefore, the product of this resistance measure and the (angular) acceleration should be torque by Newton's Second Law.

In the real world, torque is important when using hand and power tools such as drills, wrenches, and lathes. For example, it is important to know that a longer handle will have more torque and therefore will be easier to use.

Rolling Motion
Up to this point, most motion has been in the form of sliding or, in the case of projectiles, flying. Now we will introduce an entirely new for of motion which is essential for everyday life in so many ways: rolling. When an object rolls, it has two forms of movement. The first is the very familiar //translational motion,// which uses the linear variables of //x//, //v//, and //a//. An important change to note is that translational motion must be measured as the movement of the //center of mass// of the object. This is very important when slipping is involved. Consider a tire skidding on ice: the tire may be spinning very quickly, but because there is no traction, the car (or the center of mass of the tire) does not move forward at all. The second form of movement is //rotational movement//. When these two movements are combined, the result is rolling motion. Everything from a tire on pavement to a bicycle wheel going downhill to a bowling ball going down a lane is an example of rolling motion. Most examples involve rolling //without slipping//. This type of rolling exists when a given point on the outer edge of the rolling object comes in contact with a surface with static friction and propels the center of mass forward the same distance it travels on the surface. Refer to the image to the left (assume it is rolling without slipping). Picture this sphere rotating around point //P//. For the instant that //P// is in contact with the ground, this is the best way to describe the motion. As the sphere rolls, every instant a new point on the outer edge becomes the axis of rotation. If these ideal conditions continue, the center of mass will move forward at an easily predicted speed. Any object experiencing rolling without slipping will follow the equation ** vcm = ωr **. In the time that it takes point //P// to make one complete revolution, the center of mass will be propelled forward the distance of the length of the circumference. Therefore, any given point on the outer edge of an object has the same //linear// velocity as the center of mass of the object. Rarely, an object will roll with slipping. This occurs when an object rolls (rotates and moves translationally) but the velocity of the center of mass does not equal the angular velocity times the radius. Most times, the object will eventually begin to roll without slipping. In order for this change to occur, kinetic friction must be present. When the velocity of the center of mass begins larger than the angular velocity times the radius (meaning the velocity of the center of mass will drop), then the object begins sliding, without any spin. If the velocity of the center of mass begins smaller than the angular velocity times the radius, then the object begins rolling with a top spin.

When an object begins to roll, it uses rotational kinetic energy in addition to translational kinetic energy. Similar to all other rotation formulas, simply take the linear equation for kinetic energy and substitute all rotational equivalents. The kinetic energy of rolling objects follows this formula: ** K­E = ½ m v cm 2 + ½ I cm ω 2 **



This shows that for any object rolling without slipping down an incline, the linear acceleration of the center of mass is equal to **gsinθ/(1+//B//)** where //B// is the coefficient of the moment of inertia of the object.

Lab
__Purpose__ To determine the effects of varying moments of inertia as well as the accuracy of the given acceleration for center of mass equation (derived above). __Procedure__ 1. Set up an inclined plane with a motion detector at the top as show in the image on the lab 2. Measure the angle marked //θ// in the image (between the ground and the bottom of the inclined plane) 3. Adjust the motion detector settings to measure velocity versus time. 4. Hold a spherical hoop (I//=//Mr 2 ) at the top of the ramp, start recording from the motion detector, and release the hoop. 5. With LoggerPro, find the slope of the linear regression line for velocity over time. This is your measured acceleration. 6. Run four duplicate tests (steps 3-5), with slightly different angles of the inclined plane to start. Make sure the angle is not too steep as this lab only works when rolling without slipping is present. 7. Repeat steps 3-6 for a solid disk of comparable mass (I= 1 / 2 Mr 2 ) 8. For both disks, plot above acceleration versus sin//θ//. 9. With LoggerPro, find the slope of the linear regression line for both of these graphs. 10. This value should be equal to g/(1+//B//) as per the given equation above. 11. Calculate the percent error between the values.

__Data__ __**Our Results**__ This graph shows the best fit line for the acceleration (Y) vs. sin//θ// (X) for the hoop. The slope of this line //m// is 5.131. The calculated //m// value, equal to g/(1+//1//), is 4.9. The percent error between these two values is 4.7%.

This graph shows the best fit line for the acceleration (Y) vs. sin//θ// (X) for the disk. The slope of this line //m// is 6.701. The calculated //m// value, equal to g/(1+//.5//), is 6.533. The percent error between these two values is 2.6%.

In conclusion, it is clear by the very linear relationship (r=) of the acceleration versus sin<span style="font-family: 'ＭＳ 明朝';">θ that the relationship acm=(gsin<span style="font-family: 'ＭＳ 明朝';">θ )/(1+// B) // is true. This makes sense, as the force down the ramp on the wheel is mgsin <span style="font-family: 'ＭＳ 明朝';">θ. With this in mind, the acm would be gsin<span style="font-family: 'ＭＳ 明朝';">θ. That said, because the wheels have different properties (different I values) they will roll at different rates, with the lower I going faster.

Problems


[|Solution 1.jpeg]

media type="custom" key="9523620" media type="custom" key="9538590" [|Solution_2.jpeg] [|Solution_3.jpeg]