Homework #2: Vector Exercises
Michael Fowler, UVa
1. Displacement of a point object from a fixed position (which we call the origin) can be represented by a straight line from the origin to the object. We add an arrowhead on the object end of the line, and call this a "vector", the Latin word for carrier, since the line represents the path the object must be "carried" along to get it from the origin to where it is. Call this vector A. Now suppose we move the object through a distance and direction represented by a vector B.
(a) Explain with a figure why the new position of the object relative to the origin is correctly given by the vector A + B, where "+" means putting the tail of B to the head of A, and drawing the new vector A + B from the tail of A to the head of B.
(b) Draw the vector -B, and explain why it's sensible to call it that. Now draw the vector A - B. Draw A - 2B.
(c) Show that vectors of the form nA + mB, where n, m = 0, 1, 2, … form a pattern.
(d) Do the same for three vectors A, B, C that do not lie in a plane, that is, describe the series of points nA + mB + pC, where n, m, p are integers.
(e) How does the pattern in (c) change if n, m include negative integers? What if they include half-integers, that is, 0, 0.5, 1, 1.5, …?
2. Velocity can be defined as rate of change of position (or displacement).
(a) Explain why the fact that displacement is a vector implies that velocity is also a vector, that is, two velocities add vectorially to give the total velocity.
(b) Give an everyday example of velocity addition, where the velocities being added are not parallel.
3. Acceleration is rate of change of velocity.
(a) Does this mean that accelerations also add vectorially?
(b) Imagine a racing car is losing speed as it goes around a bend. Draw a diagram showing the car on the road and the direction of its acceleration.
4. (a) A force has magnitude and direction. Does it necessarily follow that the effect of two forces acting on the same point object is the vector sum of the two forces? Or is this something we have to establish experimentally?
(b) Consider a one kilogram mass hung from the ceiling using light string. Show in a diagram the forces on the mass.
(c) Show in another diagram the forces on the string.
(d) Now suppose a piece of light horizontal string is attached to the top of the mass, and slowly pulled sideways until the string from the ceiling is at 45 degrees. What is the tension in the horizontal string? In the string from the ceiling?
5. Back to the racing car losing speed as it goes around a bend. Suppose the car has a mass of 1000kg, the road is part of a circle of radius 150 meters, the car is moving at 30 meters per second, slowing down at a rate of 2 meters per second per second. What is the magnitude of the total force exerted by the road on the car? (There are three components!)
6. A cannon fires a cannonball at 50 meters per second at 45 degrees to the horizontal.
(a) First, imagine gravity has been switched off. Draw vectors showing the position of the cannonball after 0, 1, 2, 3, 4 seconds.
(b) Now let's include gravity. This will add the "natural downward acceleration" to the path you have drawn in (a). Add the appropriate vertical vectors for 1, 2, 3, 4 seconds to the displacement vectors you already drew in (a), and use this to plot the path. (Don't do any fancy calculations!)