Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Next revisionBoth sides next revision | ||
183_notes:torque [2014/11/20 10:45] – [Sign of the Torque comes from the Right-Hand Rule] caballero | 183_notes:torque [2021/05/08 18:49] – [Torques Cause Changes in Rotation] stumptyl | ||
---|---|---|---|
Line 1: | Line 1: | ||
+ | Section 5.4 and 11.4 in Matter and Interactions (4th edition) | ||
+ | |||
===== Torques Cause Changes in Rotation ===== | ===== Torques Cause Changes in Rotation ===== | ||
- | Until now, you have worked with forces and work to explain and predict | + | Until now, you have worked with [[183_notes: |
+ | ==== Lecture Video ==== | ||
+ | |||
+ | {{youtube> | ||
==== Torque ==== | ==== Torque ==== | ||
Line 15: | Line 20: | ||
$$\vec{\tau}_{A} = \vec{r}_A \times \vec{F}$$ | $$\vec{\tau}_{A} = \vec{r}_A \times \vec{F}$$ | ||
- | The //torque is a vector//; it has both a magnitude and direction. This is the first physical quantity that you have seen that seen that depends on the cross (vector) product between two vectors. It's worth taking some time to understand how this mathematics works. | + | The //torque is a vector//; it has both a magnitude and direction. The units of torque are Newton-meters ($\mathrm{Nm}$). This is the first physical quantity that you have seen that seen that depends on the cross (vector) product between two vectors. It's worth taking some time to understand how this mathematics works. |
=== The magnitude of the torque === | === The magnitude of the torque === | ||
Line 37: | Line 42: | ||
=== The direction of the torque === | === The direction of the torque === | ||
- | Torque is a vector quantity; it has a direction. How can you determine that direction? In general, this is done mathematically by [[183_notes: | + | Torque is a vector quantity; it has a direction. How can you determine that direction? |
+ | |||
+ | In general, this is done mathematically by [[183_notes: | ||
Here's the proof of that. Consider a lever arm in the $x-y$ plane that locates a force with respect to some rotation axis: $\vec{r}_A = \langle r_x, r_y, 0\rangle$ and a force that similar lives in the $x-y$ plane: $\vec{F} = \langle F_x, F_y, 0\rangle$. You want to take the cross product to find the torque: | Here's the proof of that. Consider a lever arm in the $x-y$ plane that locates a force with respect to some rotation axis: $\vec{r}_A = \langle r_x, r_y, 0\rangle$ and a force that similar lives in the $x-y$ plane: $\vec{F} = \langle F_x, F_y, 0\rangle$. You want to take the cross product to find the torque: | ||
Line 52: | Line 59: | ||
$$\vec{\tau}_A = \langle 0, | $$\vec{\tau}_A = \langle 0, | ||
+ | |||
+ | Notice that if the applied force and the lever arm point in the same (or precisely opposite directions) the torque is zero. The cross product of two parallel or anti-parallel vectors is zero. | ||
==== Sign of the Torque comes from the Right-Hand Rule ==== | ==== Sign of the Torque comes from the Right-Hand Rule ==== | ||
Line 62: | Line 71: | ||
The language often used to describe the torque direction is //into the page// and //out of the page//. It is the nature of cross products that the resultant is perpendicular to the plane defined by the vectors in the cross product. So, if the lever arm and the force vector appear on page (or screen), the torque will point into or out of the page (or screen). In the cases shown to the right, it means the vector associated with the torque points into the screen (negative, clockwise rotation) or out of the screen (positive, counter-clockwise rotation). | The language often used to describe the torque direction is //into the page// and //out of the page//. It is the nature of cross products that the resultant is perpendicular to the plane defined by the vectors in the cross product. So, if the lever arm and the force vector appear on page (or screen), the torque will point into or out of the page (or screen). In the cases shown to the right, it means the vector associated with the torque points into the screen (negative, clockwise rotation) or out of the screen (positive, counter-clockwise rotation). | ||
+ | |||
==== The Net Torque Causes Changes in Rotation ==== | ==== The Net Torque Causes Changes in Rotation ==== | ||
- | Just like you read for forces, there can be multiple torques applied to an object. That is, there might be forces applied at different locations | + | Just like you read for forces, there can be multiple torques applied to an object. That is, there might be forces applied at different locations |
$$\vec{\tau}_{net} = \sum_i \vec{\tau}_i$$ | $$\vec{\tau}_{net} = \sum_i \vec{\tau}_i$$ | ||
- | The sign of each torque is incredibly important for determining the net torque. It is the net torque that causes changes in rotation, just like it is the net force that causes changes in translation. | + | The sign of each torque is incredibly important for determining the net torque. It is the net torque that causes changes in rotation, just like it is the [[183_notes: |
+ | |||
+ | ==== Examples ==== | ||
+ | * [[: |