Right hand rule first and second rule, applications, exercises

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Abraham McLaughlin
Right hand rule first and second rule, applications, exercises

The right hand rule It is a mnemonic resource to establish the direction and the sense of the vector resulting from a vector product or cross product. It is widely used in physics, since there are important vector quantities that are the result of a vector product. Such is the case of torque, magnetic force, angular momentum and magnetic moment, for example.

Figure 1. Right hand ruler. Source: Wikimedia Commons. Acdx [CC BY-SA (http://creativecommons.org/licenses/by-sa/3.0/)].

Let be two generic vectors to Y b whose cross product is to x b. The module of such a vector is:

to x b = a.b. in α

Where α is the minimum angle between to Y b, while a and b represent its modules. To distinguish the vectors of their modules, bold letters are used.

Now we need to know the direction and the sense of this vector, so it is convenient to have a reference system with the three directions of space (figure 1 right). The unit vectors i, j Y k they point respectively towards the reader (off the page), to the right and up.

In the example in Figure 1 left, the vector to heads to the left (direction Y negative and index finger of the right hand) and the vector b goes to the reader (direction x positive, middle finger of the right hand).

The resulting vector to x b has the direction of the thumb, up in the direction z positive.

Article index

  • 1 Second rule of the right hand
    • 1.1 Alternative rule of the right hand
  • 2 Applications
    • 2.1 Angular velocity and acceleration
    • 2.2 The angular momentum
  • 3 Exercises
    • 3.1 - Exercise 1
    • 3.2 - Exercise 2
  • 4 References

Second rule of the right hand

This rule, also called right thumb rule, It is used a lot when there are magnitudes whose direction and direction are rotating, such as the magnetic field B produced by a thin, straight wire that carries a current.

In this case the magnetic field lines are concentric circles with the wire, and the direction of rotation is obtained with this rule in the following way: the right thumb points the direction of the current and the four remaining fingers are curved in the direction of countryside. We illustrate the concept in figure 2.

Figure 2. Right thumb rule to determine the direction of the magnetic field circulation. Source: Wikimedia Commons. https://upload.wikimedia.org/wikipedia/commons/c/c0/V-1_right_hand_thumb_rule.gif.

Alternative right hand rule

The figure below shows an alternate form of the right-hand rule. The vectors that appear in the illustration are:

-Speed v of a point charge q.

-Magnetic field B within which the load moves.

-FB the force that the magnetic field exerts on the charge.

Figure 3. Alternative rule of the right hand. Source: Wikimedia Commons. Experticuis [CC BY-SA (https://creativecommons.org/licenses/by-sa/4.0)]

The equation for the magnetic force is FB = qv x B and the rule of the right hand to know the direction and the sense of FB is applied like this: the thumb points according to v, the four remaining fingers are placed according to field B. Then FB is a vector that comes out of the palm of the hand, perpendicular to it, as if it were pushing the load.

Note that FB I would point in the opposite direction if the charge q was negative, since the vector product is not commutative. In fact:

to x b = - b x to

Applications

The right hand rule can be applied for various physical quantities, let's know some of them:

Angular velocity and acceleration

Both angular velocity ω as angular acceleration α they are vectors. If an object is rotating around a fixed axis, it is possible to assign the direction and sense of these vectors using the right-hand rule: the four fingers are curled following the rotation and the thumb immediately offers the direction and sense of angular velocity ω.

For its part, the angular acceleration α will have the same address as ω, but its meaning depends on whether ω increases or decreases in magnitude over time. In the first case, both have the same direction and sense, but in the second they will have opposite directions..

Figure 4. The right thumb rule applied to a rotating object to determine the direction and sense of angular velocity. Source: Serway, R. Physics.

Angular momentum

The angular momentum vector LOR of a particle that rotates around a certain axis O is defined as the vector product of its instantaneous position vector r  and the linear momentum p:

L = r x p

The rule of the right hand is applied in this way: the index finger is placed in the same direction and sense of r, the middle finger on that of p, both on a horizontal plane, as in the figure. The thumb is automatically extended vertically upwards indicating the direction and the sense of angular momentum LOR.

Figure 5. The angular momentum vector. Source: Wikimedia Commons.

Training

- Exercise 1

The top in figure 6 is spinning rapidly with angular velocity ω and its axis of symmetry rotates more slowly around the vertical axis z. This movement is called precession. Describe the forces acting on the spinning top and the effect they produce.

Figure 6. Spinning top. Source: Wikimedia Commons.

Solution

The forces acting on the top are normal N, applied on the fulcrum with the ground O plus the weight Mg, applied at the center of mass CM, with g the acceleration vector of gravity, directed vertically downwards (see figure 7).

Both forces are balanced, therefore the top does not move. However the weight produces a torque or torque τ net with respect to point O, given by:

τOR = rOR x F, with F = Mg.

What r and Mg they are always in the same plane as the top rotates, according to the right hand rule the torque τOR is always located on the plane xy, perpendicular to both r What g.

Note that N does not produce a torque with respect to O, because its vector r with respect to O is null. This torque produces a change in angular momentum that causes the spinning top to precede around the Z axis..

Figure 7. Forces acting on the top and its angular momentum vector. Left figure source: Serway, R. Physics for Science and Engineering.

- Exercise 2

Indicate the direction and the sense of the angular momentum vector L of the top of figure 6.

Solution

Any point on the top has mass mi, velocity vi and position vector ri, when it rotates around the z axis. Angular momentum Li of said particle is:

Li = ri x pi = ri x mivi

Given the ri Y vi are perpendicular, the magnitude of L it is:

Li = mirivi

Linear velocity v is related to the angular velocity ω through:

vi = riω

Therefore:

Li = miri (riω) = miritwoω

The total angular momentum of the spinning top L is the sum of the angular momentum of each particle:

L = (∑miritwo ) ω

∑ miritwo  is the moment of inertia I of the top, then:

L= Iω

Therefore L Y ω have the same direction and sense, as shown in figure 7.

References

  1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill. 
  2. Bedford, 2000. A. Engineering Mechanics: Statics. Addison wesley. 
  3. Kirkpatrick, L. 2007. Physics: A Look at the World. 6th abridged edition. Cengage Learning.
  4. Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Pearson. 
  5. Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 1 and 2. 7th. Ed. Cengage Learning.

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