RIGHT HAND RULE: FIRST AND SECOND RULE, APPLICATIONS, EXERCISES - PHYSICAL - 2025

The right-hand rule is a mnemonic to establish the direction and sense of the vector resulting from a cross 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.

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

a x b = absen α

Where α is the minimum angle between a and b, while a and b represent their 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 and k point respectively towards the reader (off the page), to the right and upwards.

In the example in Figure 1 left, vector a is directed to the left (negative y direction and right hand index finger) and vector b goes towards the reader (positive x direction, right middle finger).

The resulting vector a x b has the thumb direction, upward in the positive z direction.

Second rule of the right hand

This rule, also called the right thumb rule, is widely used when there are magnitudes whose direction and direction are rotating, such as the magnetic field B produced by a thin, rectilinear 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 remaining four fingers bend in the direction of the countryside. We illustrate the concept in Figure 2.

Figure 2. Rule of the right thumb to determine the direction of the magnetic field circulation. Source: Wikimedia Commons.

Alternative right hand rule

The following figure shows an alternative form of the right hand rule. The vectors that appear in the illustration are:

-The speed v of a point charge q.

-The magnetic field B within which the charge moves.

- F _B the force that the magnetic field exerts on the charge.

Figure 3. Alternative rule of the right hand. Source: Wikimedia Commons. Experticuis

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

Note that F _B would point in the opposite direction if the charge q were negative, since the vector product is not commutative. In fact:

a x b = - b x a

Applications

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

Angular velocity and acceleration

Both the angular velocity ω and the angular acceleration α 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 the angular velocity ω.

For its part, the angular acceleration α will have the same direction as ω, but its direction depends on whether ω increases or decreases in magnitude with 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 L _O of a particle rotating 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 in that of p, both on a horizontal plane, as in the figure. The thumb is automatically extended vertically upwards indicating the direction and sense of angular momentum L _O.

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

Exercises

- Exercise 1

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

Figure 6. Spinning top. Source: Wikimedia Commons.

Solution

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

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

τ _O = r _O x F, with F = M g.

Since r and M g are always in the same plane as the top rotates, according to the right hand rule the torque τ _O is always located in the xy plane, perpendicular to both r and g.

Note that N does not produce a torque about O, because its vector r with respect to O is zero. That torque produces a change in angular momentum that causes the top to precession 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 sense of the angular momentum vector L of the top in figure 6.

Solution

Any point on the top has mass m _i, velocity v _i, and position vector r _i, when it rotates around the z axis. The angular momentum L _i of said particle is:

L _i = r _i x p _i = r _i xm _i v _i

Since r _i and v _i are perpendicular, the magnitude of L is:

L _i = m _i r _i v _i

The linear velocity v is related to that of the angular velocity ω by:

v _i = r _i ω

Thus:

L _i = m _i r _i (r _i ω) = m _i r _i ² ω

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

L = (∑m _i r _i ²) ω

∑ m _i r _i ² is the moment of inertia I of the top, then:

L = I ω

Therefore L and 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.