Chapter 1: Describing the universe 1. 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.
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 www. 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.net 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.net 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 www.net Since the components of the vector remain unchanged, this vector must lie along the rotation axis. 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: www.net and the new components of are Let's check that the matrix we found actually does this: as required. Now let Then in the new system, the components of are: and so Since the initial velocity is the particle's velocity at time is: and the path is intially parabolic: www.net This result is valid so long as the initial velocity has not changed appreciably, so that the acceleration is approximately constant.
That is: or times (the cyclotron period divided by. The time may be quite long if is small. Now we convert back to the original coordinates: 5. A solid body rotates with angular velocity Using cylindrical coordinates with axis along the rotation axis, find the components of the velocity vector at an arbitrary point within the body.
Use the expression for curl in cylindrical coordinates to evaluate Comment on your answer. The velocity has only a component. Then the curl is given by: Thus the curl of the velocity equals twice the angular velocity- this seems logical for an operator called curl. Starting from conservation of mass in a fixed volume use the divergence theorem to derive the continuity equation for fluid flow: where is the fluid density and its velocity.
The mass inside the volume can change only if fluid flows in or out across the boundary. Thus: where flow outward ( decreases the mass. Now if the volume is fixed, then: Then from the divergence theorem: and since this must be true for any volume then 7. Find the matrix that represents the transformation obtained by (a) rotating about the axis by 45 counterclockwise, and then (b) rotating about the axis by 30 clockwise.
What are the components of a unit vector along the original axis in the new (double- prime) system? The first rotation is represented by the matrix The second rotation is: And the result of the two rotations is: www.net The new components of the orignal axis are: 8. Does the matrix represent a rotation of the coordinate axes? If not, what transformation does it represent? Draw a diagram showing the old and new coordinate axes, and comment. The determinant of this matrix is: Thus this transformation cannot be a rotation since a rotation matrix has determinant Let's see where the axes go: and www.net while These are the components of the original and axes in the new system. The new and axes have the following components in the original system: where Thus: The picture looks like this: Problem 8: www.net The matrix represents a reflection of the and axes about the line 9.
Represent the following transformation using a matrix: (a) a rotation about the axis through an angle followed by (b) a reflection in the line through the origin and in the -plane, at an angle 2 to the original axis, where both angles are measured counter-clockwise from the positive axis. Express your answer as a single matrix. You should be able to recognize the matrix either as a rotation about the axis through an angle or as a reflection in a line through the origin at an angle to the axis. Decide whether this transformation is a reflection or a rotation, and give the value of (Note: For the purposes of this problem, reflection in a line in the plane leaves the axis unchanged.) Since only the and components are transformed, we may work with matrices.
The rotation matrix is: The line in which we reflect is at 2 to the original axis and thus at to the new axis. Thus the matrix we want is (see Problem 8 above): Thus the complete transformation is described by the matrix: The determinant of this matrix is , and so the transformation is a reflection. It sends to and to so it is a reflection in the axis ( 10. Using polar coordinates, write the components of the position vectors of two points in a plane: with coordinates and and with coordinates and (That is, write each vector in the form What are the coordinates and of the point whose position vector is www.net Hint: Start by drawing the position vectors.
Problem 10 The position vector has only a single component: the component. Thus the vectors are: and The sum also only has a single component: where, from the diagram , and: Thus has coordinates where and thus We can check this in the special case Then as required.net This document created by Scientific WorkPlace 4. A skew (non-orthogonal) coordinate system in a plane has axis along the axis and axis at an angle to the axis, where (a) Write the transformation matrix that transforms vector components from the Cartesian system to the skew system. (b) Write an expression for the distance between two neighboring points in the skew system.
Comment on the differences between your expression and the standard Cartesian expression. (c) Write the equation for a circle of radius with center at the origin, in the skew system.11 (a) The new coordinates are: and Thus the transformation matrix is: Compare this result with equation 1. Here the components are given by www.net (b) The cross term indicates that the system is not orthogonal. We could also have obtained this result from the cosine rule.
(c) The circle is described by the equation a result that could also be obtained by applying the cosine rule to find the radius of the circle in terms of the coordinates and 12. Prove the Jacobi identity: The triple cross product is and thus Since the dot product is commutative, the result is zero, as required. Evaluate the vector product in terms of triple scalar products. What is the result if all four vectors lie in a single plane? What is the result if and are mutually perpendicular? What is the result if www.net We can start with the bac-cab rule: Equivalently, we may write: If all four vectors lie in a single plane, then each of the triple scalar products is zero, and therefore the final result is also zero.
If and are mutually perpendicular where the plus sign applies if the vectors form a right-handed set, and If then and 14. Evaluate the product in terms of dot products of and www. Use the vector cross product to express the area of a triangle in three different ways. Hence prove the sine rule: First we define the vectors and that lie along the sides of the triangle, as shown in the diagram.
Then the area equals the magnitude of or of or of Hence Dividing through by the product we obtain the desired result. Use the dot product to prove the cosine rule for a triangle: With the vectors defined as in the diagram above, But if and lie along two sides of a triangle s shown, then the third side Thus www.net as required. A tetrahedron has its apex at the origin and its edges defined by the vectors and each of which has its tail at the origin (see figure). Defining the normal to each face to be outward from the interior of the tetrahedron, determine the total vector area of the four faces of the tetrahedron.
Find the volume of the tetrahedron.17 With direction along the outward normal, the area of one face is The total area is given by: Expanding out the last product, and using the result that : since The volume is 1/6 of the parallelopiped formed by the three vectors, (or 1/3 base times height of tetrahedron) and so 18. A sphere of unit radius is centered at the origin. Points and on the surface of the sphere have position vectors and Show that points and www.net on the sphere, located on a diameter perpendicular to the plane containing the points and have position vectors given by where is the angle between the vectors and .18 The triangle has sides given by the vectors and. The plane of the triangle may thus be described by the vector This vector is normal to the plane.
The vector is a unit vector, as are the vectors and since the sphere has unit radius. Thus we may write and Thus and thus www.net To obtain both ends of the diameter, we need to add the sign, as given in the problem statement. Show that for any scalar field because the order of the partial derivatives is irrelevant.
