of the acceleration. To get these terms, we treat the velocity components as separate vectors that can change in magnitude and direction and apply to each of them the same procedure we used on the position vector in the preceding paragraph.
Consider first the radial velocity (
Next consider the tangential velocity (
The vector sum of these components yields the same acceleration as Equation 1.20.
Exercises
Descriptions of the systems referred to in the exercises are contained in Appendix A.
1 1.1 Show that the absolute velocity of point of system 1 is,
2 1.2 Show that the absolute acceleration of point of system 1 is,(1.49)
3 1.3 Let the coordinate system shown in system 2 be fixed in and show that the acceleration of with respect to is,(1.50)
4 1.4 Let the coordinate system shown in system 2 be fixed in and repeat the previous problem. Note that the angle is constant in this coordinate system so that the vector has no rate of change of magnitude.
5 1.5 Show that the “rolls without slipping condition” in system 5 requires that the drum have an angular velocity in the counterclockwise direction.
6 1.6 Show that the absolute acceleration of the center of the drum in system 5 has a component,down the plane and a component perpendicular to the plane that can be written as,
7 1.7 Define a right handed coordinate system in system 6 where is aligned with and points upward and show that,
8 1.8 Show that the “rolls without slipping condition” in system 14 requires that the disk have an angular velocity in the clockwise direction.
9 1.9 Show that the acceleration of the center of mass of the uniform rod in system 16 has a component,aligned with the rod and a component perpendicular to the rod that can be written as,
10 1.10 Using the coordinate system of Exercise 1.7, show that the absolute angular acceleration of the massless rigid rod in system 6 is,
11 1.11 Using a coordinate system fixed in the ground with positive to the right and positive up, show that the acceleration of the center of mass of the the rod in system 12 is
12 1.12 Show that the absolute acceleration of the center of mass of the rod in system 16 is,where points from to .
13 1.13 Consider system 23. Using the rotating coordinate system shown, show that the absolute velocity of the mass is,
14 1.14 Consider system 23 again. Define a ground fixed coordinate system (,,) obtained by a plane rotation of the (,,) system through an angle in the negative direction. That is, points from to and is aligned with . Show that the position of the mass with respect to the point in this system is,Differentiate this position vector to get the absolute velocity of the mass and show that you could get the same result by transforming the result of Exercise 1.13 using a plane rotation.
15 1.15 Finally, for system 23. Define a body fixed coordinate system (,,) obtained by a plane rotation of the (,,) system through an angle in the negative direction. That is, points from to and is aligned with . The position of the mass with respect to the point in this system is,Differentiate this position vector to get the absolute velocity of the mass and show that you could get the same result by transforming the result of Exercise 1.13 using a plane rotation. Be sure to get the correct angular velocity for the coordinate system before differentiating.
Notes
1 1 The convention used here is that a vector with an overdot such as is used to represent the rate of change of magnitude of the vector and the overdot is not to be interpreted as a shorthand method of signifying the total derivative of the vector. For a scalar function there is only a magnitude and the overdot will represent its rate of change.
2 2 Let the cross product of two vectors, and , be the vector, According to the right hand rule, the direction of is the direction aligned with the thumb of your right hand if you point that hand in the direction of and curl your fingers towards .
3 3 Three dimensional sets of unit vectors shown in two dimensional figures such as Figure 1.3 will be shown with the positive sense of the vector out of the plane represented by a curved arrow using the right hand rule (e.g. in Figure 1.3).
4 4 Readers are encouraged to review the rules for cross multiplication of vectors.
5 5 Gaspard Gustave de Coriolis (1792–1843), an engineer and mathematician, introduced the terms “work” and “kinetic energy” to engineering analysis but is best remembered for showing that the laws of motion could be used in a rotating reference frame if an extra term called the Coriolis acceleration is added to the equations of motion.
