We begin an example of the application of vector functions. The first clip discusses a rotating disk, and a beam rotating independently near the rim of that disk.
A point on a disk rotating about an axis through its center moves along a circular path which can be described by a vector function r_1 ( t).
The position of a point on a beam rotating about that point on the disk can be described, relative to its axis of rotation, by a vector function r_2 (t).
The position of that second point therefore can be described by a vector function r(t) = r_1 (t) + r_2 (t).
The vector function r(t) = r_1 ( t) + r_2 (t) is depicted on a graph, and specific vector functions r_1(t) and r_2(t) are constructed to correspond to the motion of an amusement park.
The derivative funcxiton r ' (t) is easily calculated, and corresponds to the velocity function for our point. The direction of r ' (t) is the (ever-changing) direction in which you would be looking if you were on the ride, looking straight ahead. The magnitude || r ' (t) || tells us the speed. We can maximize or minimize this function using standard optimization techniques, finding the derivative of this expression and setting it equal to zero, etc.. We won't actually do this, since the expressions would be long and messy, but if necessary we could follow the rules of differentiation and obtain the desired results.
The carnival ride is fun because of the way the direction your visual field changes, and also because of the sensation of speeding up and slowing down and being tossed from one side of the seat to the other. These sensations are due to the acceleration we experience. The acceleration function is easily calculated by taking the derivative of the velocity function, obtaining r'' (t) . The expression gets longer with each differentiation, but the differentiation is straightforward.
At any instant the acceleration has a component in the direction of the velocity, easily calculated by projecting the acceleration vector onto the velocity vector, and a component perpendicular to the velocity, easily calculated by subtracting the projection from the acceleration vector itself. The acceleration component parallel to the velocity speeds us up or slows us down, while the component perpendicular to the velocity changes our direction, turning us more or less quickly toward our left or right.
We can consider the r(t) vector for a point on a rotating disk as it coasts to rest, or for a rotating disk as the axis is tilted, continuously changing the plane of motion. The could be complicated to visualize, but with vector techniques is not particularly difficult to model and analyze.
We can consider the motion of a ball rolled across a sloping plane, which turns out to be pretty easy to visualize and model.
We can consider the motion of a point on a rotating washer as it swings, or on a coin we flip.
We now consider some simpler vector functions. The function r ( t ) = sqrt(t) i + 1/t j describes an easily-understood path.
The function r ( t ) = cos(t) i + 2 sin(t) j + 1/2 t k is analyzed by first analyzing the function r_1 ( t) = cos(t) i + 2 sin(t) j, which describes an ellipse in the xy plane. We then consider the function r_2 ( t) = 1/2 t k , which describes a vector which progressively rises along the z axis. Adding these vector functions we obtain the function r (t) , which we graph in 3-dimensional space to get a good idea the corresonding path.
The functions r ' (t) and r '' (t), corresponding to our function r ( t ) = cos(t) i + 2 sin(t) j + 1/2 t k, can be interpreted as velocity and acceleration functions for the original position function, giving us insight into the dynamics of the motion of a point following this curve.
Generalizing our insights from the preceding models, we consider how vector functions F(t) and G(t), and a scalar function h(t), can be combined by operations like addition, dot product, cross product and composition. The rules of differentiation of these various combinations functions are very similar to those learned in first-semester calculus.
The derivative of a vector function is defined by a difference quotient similar to that used to define the derivative in first-semester calculus. The derivative of the vector function has at any instant a magnitude and a direction. We see why the rules we have implicitly used so far to find the derivatives of our r(t) functions are in fact valid.
The motion of the ball rolling along the tilted plane is characterized by acceleration function a ( t ) = 0 i - g_d k. This function is easily integrated to obtain the corresponding velocity function, including an arbitrary constant that allows us to fit this function to a given situation. The velocity function is itself integrated to find a position function, again with an integration constant that allows us to further adjust our result to represent a given situation.
We adjust our solution from the previous discussion to fit the initial speed v_0 and angle of elevation theta of a projectile.
We represent a typical set of initial conditions on an actual incline and see that our function does plausibly do a good job of representing the actual motion.
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A vector function F(t) can be expressed in polar coordinates (r(t), theta(t))_polar.
The change in the vector F(t) between t and t + `dt has a radial component, i.e., a component in the direction of F(t), and a component perpendicular to the radial direction.
The polar coordinate at t is r(t), which is equal to the magnitude ||F(t)|| of our vector function, and the angular coordinate is theta(t), the direction of the radial line with respect to the polar axis. It follows that F(t) = r(t) ( cos(theta(t)) i + sin(theta(t)) j ).
Keep in mind that both r(t) and theta(t) are functions of t. As t changes, we expect r and theta to both change, causing changes in our vector function in both the radial and tangential directions.
We can construct set of unit vectors appropriate to polar coordinates by defining, at every point of the curve defined by our vector function R ( t ), the unit vectors u_r = cos(theta) i + sin(theta) j, and u_theta = - sin(theta) i + cos(theta) j. At every point u_r is in the radial direction and u_theta is perpendicular to the radial direction in the right-handed sense.
As t changes our unit vectors u_r and u_theta change, according to our function theta(t). We see that (u_r) ' = u_theta * (dTheta / dt), and (u_theta) ' = - u_r (dTheta / dt ).
The vector function R(t) = r(t) * u_r is seen to have derivative R ' (t) = r ' (t) u_r + r(t) u_theta * (dTheta / dt).
We can choose whether to represent a given vector function in terms of rectangular coordinates, where R(t) = x(t) i + y(t) j and R ' (t) = x ' (t) i + y ' (t) j, or the form R(t) = r(t) * u_r with its derivative R ' (t) = r ' (t) u_r + r(t) u_theta * (dTheta / dt).
The familiar rectangular formulation might seem to be simpler, and often is, but there are cases where the polar representation is in fact simpler.
One example is that of a planet orbiting the Sun (or of any satellite orbiting a planet). In this case the acceleration function for the planet, which directly follows from Newton's Law of Universal Gravitation, is a(t) = - G M / r^2 * u_r. The force is purely radial in nature, with magnitude inversely proportional to r.
The acceleration is a(t) = R '' ( t ). We find R ''(t) by taking the derivative of R ' (t). Our result is R ''(t) = (r '' - r (dTheta/dt)^2) u_r + (2 r ' (dTheta/dt) + r (theta ''(t) ) u_theta.
Setting R ''(t) = (r '' - r (dTheta/dt)^2) u_r + (2 r ' (dTheta/dt) + r (theta ''(t) ) u_theta equal to our acceleration - G M / r^2 * u_r, we find that r '' - r (dTheta / dt)^2 = -G M / r^2, and 2 r ' (dTheta / dt) + r (theta ''(t)) = 0.
The first equation is beyond the scope of this course, requiring techniques of differential equations for its solution.
The second has a relatively simple and profound interpretation.
We solve the equation 2 r ' (dTheta / dt) + r (theta ''(t)) = 0 for r in terms of theta ' (t) = dTheta / dt.
Our solution leads us to the conclusion that r^2 (dTheta / dt) is constant. (Note that r (dTheta / dt) is the component perpendicular to the radial direction of the velocity of our planet, so that r^2 dTheta/dt is the product of r with this component. Physics students might want to note that expression, multiplied by the mass of the planet, is the magnitude of its angular momentum, which we conclude must be constant).
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We interpret the result r^2 dTheta / dt = constant in terms of the area swept out by the radial vector, obtaining the Kepler's Second Law (the second of three laws of planetary motion, empirically derived by Kepler and a fairly direct precursor to the development of classical physics).
We consider whether the ellipse x(t) = 3 cos(omega t), y(t) = 5 sin(omega t) could describe the path of a planetary orbit with the Sun at the origin. We first test this by finding the corresponding expression for r^2 dTheta/dt. Taking the somewhat messy derivative of our result we conclude that r^2 dTheta / dt is not constant for this motion.
We do find that the function R (t) = 3 cos(omega t) i + 5 cos(omega t) j does have a second derivative that is in the negative radial direction. This in itself does not contradict Newton's Law of Universal Gravitation. However the preceding result for r^2 dTheta/dt showed that this parameterization does indeed contradict that Law. We conclude that the acceleration corresponding to this parameterization is not inversely proportional to the square of r.
In terms of the this analysis we motivate the usefulness of the unit tangent and unit normal vectors.
We calculate unit tangent and unit normal vectors for our function R (t) = 3 cos(omega t) i + 5 cos(omega t) j .
Among other things the instructor raises the question of the dimension T ' (t) and seems to arrive at a contradiction, which is then resolved.
We will continue with the calculation of the unit normal vector, which requires only that we divide T ' (t) by its magnitude. Very messy and algebraically challenging, but simple in concept.
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If || R (t) || is constant, then R ' (t) is perpendicular to R(t). We argue this first in terms of the properties of isosceles triangles. We then show the same result in terms of the unit vectors u_r and u_theta.
If the position of a point is given by the vector function R(t) = r(t) u_r then the acceleration vector R ''(t) has radial component r (t) * (theta ' (t)) and tangential component r(t) * theta '' (t). The former is a centripetal acceleration of the point, the latter is the rate at which its speed changes.
We define the unit tangent and principle unit normal vectors and the components of the acceleration vector in the directions of these unit vectors.
We sketch part of the graph of R(t) = -3 sin(9 t) i + 7 cos(9 t) j + t^2 k, based on the graph of the ellipse defined by -3 sin(9 t) i + 7 cos(9 t) j.
We begin to calculate R ', T and T ' for R(t) = -3 sin(9 t) i + 7 cos(9 t) j + t^2 k
In interest of time we didn't complete the calculation of the second derivative T '.
We discuss some aspects of T ' for the preceding function.
For a given curve we consider the effect of the parameterization of the rute on the rate of change of the unit tangent vector with respect to the parameter. Different parameterizations will result in different rates of change.
If the curve is parameterized in terms of arc distance along the curve, then the corresponding derivative of the unit tangent vector gives us the curvature.
Curvature is easily calculated given a reasonably well-behaved function R (t).
When we partition the interval between t = a and t = b, and consider the arc distance corresponding to a typical interval of the partition, we conclude that the arc distance is the integral of R ' (t), between limits t = a and t = b.
We calculate the curvature and an arc length of the curve defined by `R ( t ) = A ( cos(pi t) i + sin(pi t) j).
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We now analyze the curve R(t) = A cos(pi t) i + A sin(pi t) j + t^2 k.
Warning: There appears to be an error somewhere in the calculation of the curvature, as will become apparent at the end when we analyze the behavior of the curvature for small t.
We compare our result to the previous function R(t) = A ( cos (pi t) i + sin(pi t) j), and sketch an approximate graph of both vector functions.
We consider the question of arc distance.
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