VDU_Physics_1_001.htm

## Physics 1.-    Vectors

Javier Montenegro Joo

## jmj@VirtualDynamics.Org

### Physics Virtual Lab (PVL)

The Physics Virtual Lab is a collection of at least 200 Physics simulation modules at the university level, covering areas such as Mechanics, Oscillations, Waves, Fluids, Thermodynamics, Optics, Electricity, and Magnetism. The PVL is in fact a virtual lab where the students can interactively run Physics experiments with their own data and, collect data that may be used to plotting curves of behavior, just like with any traditional physics lab. Many of the problems in this web page have been solved with the "Problem Generators" included in the PVL.

To know more about the PVL click:     http://www.virtualdynamicssoft.com

  The sketch shows four vectors  A, B, C and D.  Write their mathematical expressions. In all cases verify your result by computing the length of the vector. Solution.- (1)  Given the vector   V = -3x + 2y -7z     find: (a) Module, (b) Unit vector, (c) angles with the coordinated axes, (d) Verify that the sum of the squares of its direction cosines is one.

Solution Vector Calculator module included in the PVL

The user enters the data of the vectors and the module computes their length, angles with the axes, angle between vectors, scalar product, vector product, projections, etc. Many of the vector problems in this web page have been solved with the PVL vector calculator module.

(2) The cosines of the angles that a unit vector of a vector V makes with the Cartesian XYZ axes are respectively   0.451, 0.876 and 0.172.   It is known that the magnitude of vector V is 45 units.  (a) Construct the vector (b) Find the angles between vector V and the X, Y and Z  axes.

Solution (3)   In a XYZ-coordinated system a vector V whose magnitude is 133 units makes an angle of 165° with the X-axis and an angle of 90° with the Z-axis. Find the components of the vector and write its unit vector. In order to verify your results, compute the moduli of vectors V and its unit vector.

Solution

Since the vector makes an angle of 90° with the Z-axis, the vector is on the XY-plane and its Z-component is zero.  The Boom Crane simulation module is ideal to study torque. This module transports a random weight to a random position on the right-hand side of the crane and, the student must compute the equilibrating weight to be placed at a certain position on the other side.

(4)   In the sketch, express the vectors A, B, C, D, E and  F in terms of its cartesian components: Solution

In order to express a vector in terms of its components, start at the origin of coordinates and try to reach the tip of the vector along two different routes, one of them including the vector. (5) The unit-vector of a vector whose magnitude is 133 is given by

u = -0.965 x + 0.258 y

(a) Construct the vector  (b) Verify its length (c) Find the angles the vector makes with the xyz-coordinated axes.

Solution.- (6)  The sketch shows the route followed by a robot over the horizontal ground. Calculate the total displacement and the distance traveled by the robot. Solution.-

It can be seen in the sketch that the components of each vector are respectively: (7)  Given the vectors A and B:      A = 12 x  - 6 y  + 5 z      and        B =  7 x  - 3 z

Find the angle between both

Solution.- (8)  A vector of   9.11   units length makes the following angles with the coordinated axes X,Y and Z, respectively:   140.21o ,  70.77o  and  123.29.     Construct the vector and its unit vector.

Solution.- (9)  Given the vectors    A = 7x – 9y +12 z       and        B = -12 x + 9y -7z    Calculate: (a) their scalar product (b) their vector product (c) the angle between both vectors (d) The unit vector in the direction of their vector product.

Solution.- (10)  Given the vector   A = -7 x -5 y + 13 z, calculate:  (a) its module, (b) unit vecctor, (c) direction cosines (d) angles with the coordinated axes and, (e) vector direction angles (angles with axes x, y and z).

Solution.- (11) In the figure, calculate:  (a) the scalar product of vectors M and N (b) the angle between vectors M and N (c) the vector product of vectors M and N. (d) the unit vector in the direction of vector C = M x N (e) verify that the length of the unit vector in the direction of M x N, is one. Solution.-

In order to express the vector in terms of its components, start at the origin of coordinates and try to reach the tip of the vector along two different routes, one of them including the vector. (12)  Construct the mathematical expressions of the four unit vectors  a, b, c  and  d,  shown in the sketch. Each unit vector makes an angle of 45° with the horizontal. Solution.- (13) The figure shows four vectors A, B, C and D and their respective magnitudes, each one making an angle of 45° with the horizontal. Determine their mathematical expressions and their unit vectors. Solution.- (14) Find the resultant of two forces of 8 N and 12 N respectively, which act on the same point and make an angle of 70o between them. Determine the angle that the resultant makes with the 8N force.

Solution.-

Consider the sketch bellow. In (a) the two forces and the angle of 70° between them are shown. In (b) the F2  force has been translated without rotation to the extreme of F1 and, the resultant vector R of the two applied forces is now evident. Note that the angle of F2 with the direction of F1 has not changed. (15) The three forces shown in the sketch are in equilibrium (it may be the case of a traffic light hanging from two diagonal cables and a vertical one). Determine the three angles between the forces. Solution.-

Since the three forces are in equilibrium, their sum must be zero. Then by displacing the forces without rotation they must shape up a triangle, like the one shown in the sketch below. In order to find the unknown angles apply the Cosine Law. 