複数の方向のベクトル

Vectors are quantities with both magnitude and direction-unlike scalars, which have only a magnitude and sign.

Force, acceleration, and velocity are examples of vectors. While mass, energy, and time, are examples of scalars.

A vector is usually represented by an arrow. The arrow's length corresponds to its magnitude and the angle indicates direction.

This video will show a system of forces that can be analyzed with vector addition and subtraction, and demonstrate how such operations produce results that are important for understanding several physical phenomena.

Describing a vector requires a coordinate system. Within this chosen frame of reference, this example of a ball kicked into the air has an initial velocity vector. As explained earlier, the length of the arrow represents the magnitude of the velocity. And the direction of the vector is its angle from the ground.

Any vector may be decomposed into components, which are vectors themselves along the x- and y-axes. If the ball's initial velocity is 20 meters per second at 60 degrees, the horizontal component is speed times cosine of 60 degrees and has a magnitude of 10 meters per second. The vertical component is speed times sine of 60 degrees and has a magnitude of about 17.3 meters per second.

Vector addition of horizontal and vertical components reconstructs the original velocity vector. To add vectors, imagine placing the head of one to the tail of the other. In this example the vectors happen to be at a right angle. The sum results when traveling directly from the tail of the first to the head of the second.

These components are at a right angle, so the magnitude of the sum is given by the Pythagorean theorem. The angle is the arctangent of the vertical component divided by the horizontal component.

When adding two vectors that are not perpendicular, decompose each into x- and y-components then add corresponding components. Finally, calculate the vector sum of horizontal and vertical components as explained before. Subtracting one vector from another is equivalent to negating the second vector and adding it to the first. As before, decompose each vector into x- and y-components. Then subtract the smaller x-component from the larger one and do the same for y-components. Then, same as before, calculate the vector sum of the resulting x- and y-components.

To demonstrate the addition and subtraction of vectors in a physics lab, the equipment commonly used is a force table. This is a disk with angles marked around the perimeter, a ring in the center attached to cords with masses at the other end suspended by pulleys. The masses produce forces, which are the vectors to be studied. The force along each cord is equal to the gravitational force, or mg, with units of Newtons.

Now, in this set-up, if there are just two equal masses at 180 degrees from each other, then they produce forces with a vector sum of zero. This condition is called equilibrium, which results in zero acceleration and thus the ring won't move.

But if the two forces pulling the ring do not cancel each other, for example due to change in angle, then the non-zero net force would cause the ring to move. In such instances, if we know the magnitudes and directions of these forces, then we can use vector addition and subtraction to calculate the third force needed to re-establish equilibrium.

In the next section we will show how to conduct such force table experiments that test the theoretical principles of vector addition and subtraction

If the two forces are equal and opposite, the ring at the center of the table should not move. In this case, each force vector exactly opposes the other in magnitude and direction. The vector sum has zero magnitude, which is the condition of zero net force, or equilibrium.

To validate the principles of vector addition and subtraction, set up the masses and angles for forces A and B as indicated on the first line of this table. Keep the angle for A at zero degrees. Now, set up the third force by adding masses and changing the angle until the ring does not move.

After achieving equilibrium, calculate the force of C by multiplying its mass by the acceleration due to gravity. Also, record the magnitude and angle for force C.

Repeat this test for the three different cases and record the magnitude and angle of force C each time.

For the four experimental set-ups, this table shows the calculated magnitudes of forces A and B, and angles of B with respect to A. Using the first set-up as an example, we can calculate force C needed to establish equilibrium on the table.

Here force A has a magnitude of 0.98 Newtons at 0°. Force B has the same magnitude of 0.98 Newtons but an angle of 20°. To determine the vector for C, decompose forces A and B into their x- and y-components. Note force A is directed only along the x-axis and has no y-component. Then add the components to yield the x- and y-vectors, which are the sum of A and B vectors.

To achieve equilibrium, the x- and y-components of C must be the opposite of these vectors. To obtain the vector C, move the tail of its y-component to the head of the x-component. Then add the two vectors using the Pythagorean theorem to find the magnitude of vector C. And the angle for C is the arctangent of the vertical component divided by the horizontal component. Therefore, the calculated magnitude of C turns out to be 1.93 Newtons at an angle of 10° with respect to the x-axis.

Now during the experiment, we calculate C through observation and trial and error, by adjusting the weights and angles to prevent motion of the ring on the force table.

And this table shows that experimental and calculated results for both magnitude and angle match closely for all four setups. This agreement validates the representation of forces as vectors. The difference may be attributed to limitations in the accuracy of the weights, measurement accuracy of the angle and unaccounted forces caused by friction on the force table and with the pulleys.

Vector addition and subtraction are used in both simple and complex applications. Let's take a look at some of them.

When touring a city like New York, the distance is usually measured in blocks, and the directions are north, south, east, and west.

A person walking four blocks east and three blocks north undergoes a change in position, which is a vector quantity. Therefore, by applying the equations for vector addition, one can calculate the magnitude and direction of the vector between the start and end points of the walk.

From walking to flying: a pilot is constantly performing mental vector addition and subtraction to maneuver the airplane. By using the wings' flaps and ailerons, a pilot can adjust lift against gravity. If lift is greater than gravitational force, the plane ascends. If lift is less than gravitational force, it descends.

Similarly, a pilot uses the engines to adjust thrust against drag. If thrust is greater than drag, the plane accelerates. If thrust is less than drag, it decelerates.

When the sum of these four forces equals zero, the plane is in equilibrium and cruises at constant speed and altitude.

You've just watched JoVE's introduction to vectors. You should now know how to add and subtract vectors, and understand how certain physical quantities behave as vectors. Thanks for watching!