Mathematics_for_Physicists-Lea-solutions.pdf

(17058 KB) Pobierz
584685760 UNPDF
Chapter 1: Describing the universe
1 . Circular motion . A particle is moving around a circle with angular velocity Write its
velocity vector as a vector product of and the position vector with respect to the
center of the circle. Justify your expression. Differentiate your relation, and hence derive the
angular form of Newton's second law (
from the standard form (equation 1.8).
The direction of the velocity is perpendicular to and also to the radius vector and is
given by putting your right thumb along the vector : your fingers then curl in the direction
of the velocity. The speed is
Thus the vector relation we want is:
Differentiating, we get:
since is perpendicular to The second term is the usual centripetal term. Then
and
since is perpendicular to and for a particle
584685760.051.png 584685760.054.png 584685760.055.png 584685760.056.png 584685760.001.png 584685760.002.png 584685760.003.png 584685760.004.png 584685760.005.png 584685760.006.png 584685760.007.png
2 . Find two vectors, each perpendicular to the vector
and perpendicular to each
other. Hint: Use dot and cross products. Determine the transformation matrix that allows
you to transform to a new coordinate system with
axis along and
and
axes
along your other two vectors.
We can find a vector perpendicular to by requiring that
A vector satsifying
this is:
Now to find the third vector we choose
To find the transformation matrix, first we find the magnitude of each vector and the
corresponding unit vectors:
and
The elements of the transformation matrix are given by the dot products of the unit vectors
along the old and new axes (equation 1.21)
To check, we evaluate:
as required. Similarly
584685760.008.png 584685760.009.png 584685760.010.png 584685760.011.png 584685760.012.png 584685760.013.png 584685760.014.png 584685760.015.png 584685760.016.png 584685760.017.png
and finally:
3 . Show that the vectors (15, 12, 16), (-20, 9, 12) and (0,-4, 3) are mutually
orthogonal and right handed. Determine the transformation matrix that transforms from the
original
cordinate system, to a system with
axis along
axis along and
axis along Apply the transformation to find components of the vectors
and
in the prime system. Discuss the result for vector
Two vectors are orthogonal if their dot product is zero.
and
Finally
So the vectors are mutually orthogonal. In addition
So the vectors form a right-handed set.
To find the transformation matrix, first we find the magnitude of each vector and the
corresponding unit vectors.
So
584685760.018.png 584685760.019.png 584685760.020.png 584685760.021.png 584685760.022.png 584685760.023.png 584685760.024.png 584685760.025.png 584685760.026.png 584685760.027.png 584685760.028.png 584685760.029.png
Similarly
and
The elements of the transformation matrix are given by the dot products of the unit vectors
along the old and new axes (equation 1.21)
Thus the matrix is:
Check:
as required.
Then:
and
584685760.030.png 584685760.031.png 584685760.032.png 584685760.033.png 584685760.034.png 584685760.035.png 584685760.036.png 584685760.037.png
Since the components of the vector remain unchanged, this vector must lie along the
rotation axis.
4 . A particle moves under the influence of electric and magnetic fields and Show that
a particle moving with initial velocity
is not accelerated if is perpendicular to
A particle reaches the origin with a velocity
where is a unit vector in the
direction of and
If
and
set up a new
coordinate system with
axis along
and
axis along Determine the
particle's position after a short time Determine the components of
and
in both
the original and the new system. Give a criterion for ``short time''.
But if is perpendicular to then
so:
and if there is no force, then the particle does not accelerate.
With the given vectors for and then
Then , since
Now we want to create a new coordinate system with
axis along the direction of
Then we can put the -axis along and the axis along The components in the
original system of unit vectors along the new axes are the rows of the transformation matrix.
Thus the transformation matrix is:
584685760.038.png 584685760.039.png 584685760.040.png 584685760.041.png 584685760.042.png 584685760.043.png 584685760.044.png 584685760.045.png 584685760.046.png 584685760.047.png 584685760.048.png 584685760.049.png 584685760.050.png 584685760.052.png 584685760.053.png

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin