1. Solving Problems

2. [1] The position of a particle moving under uniform acceleration is some function of time and the acceleration. Suppose we write this position s = kam tn where , k is a dimensionless constant. Show by dimensional analysis that this expression is satisfied if if m = 1 and n = 2
Solution
and

3. [2] Newton’s law of universal gravitation is represented by
Here F is the magnitude of the gravitational force exerted by one small object on another , M and m are the masses of the objects, and r is a distance. Force has the SI units kg ·m/ s2. What are the SI units of the proportionality constant G?
Solution:

4. [3] A solid piece of lead has a mass of 23. 94 g and a volume of 2
[3] A solid piece of lead has a mass of g and a volume of 2.10 cm3. From these data, calculate the density of lead in SI units (kg/ m3).
Solution
One centimeter (cm) equals 0.01 m.
One kilometer (km) equals 1000 m.
One inch equals 2.54 cm
One foot equals 30 cm…
Example:

5. [4] If the rectangular coordinates of a point are given by (2, y) and its polar coordinates are ( r , 30°), determine y and r.
Solution:
then
then
[4] Two points in the xy plane have Cartesian coordinates (2.00, -4.00) m and ( -3.00, 3.00) m. Determine (a) the distance between these points and (b) their polar coordinates. a)
Solution:

6. For (2,-4) the polar coordinate is (2,2√5) sinceb
For (2,-4) the polar coordinate is (2,2√5) since

7. [5] Vector A has a magnitude of 8. 00 units and makes an angle of 45
[5] Vector A has a magnitude of 8.00 units and makes an angle of 45.0 ° with the positive x axis. Vector B also has a magnitude of 8.00 units and is directed along the negative x axis. Using graphical methods, find (a) the vector sum A + B and (b) the vector difference A - B.
Solution:

8. [6] Given the vectors A = 2. 00 i +6. 00 j and B = 3. 00 i - 2
[6] Given the vectors A = 2.00 i j and B = 3.00 i j, (a) draw the vector sum , C = A + B and the vector difference D = A - B. (b) Calculate C and D, first in terms of unit vectors and then in terms of polar coordinates, with angles measured with respect to the , +x axis.
Solution:

9. [7] Consider the two vectors A = 3 i - 2 j and B = i - 4 j
[7] Consider the two vectors A = 3 i - 2 j and B = i - 4 j. Calculate (a) A + B, (b) A - B, (c) │A + B│, (d) │A - B│, and (e) the directions of A + B and A - B.
Solution:
Direction of A+B
Direction of A-B

10. [8] The vector A has x, y, and , z components of 8. 00, 12. 0, and -4
[8] The vector A has x, y, and , z components of 8.00, 12.0, and units, respectively. (a) Write a vector expression for A in unit vector notation. (b) Obtain a unit vector expression for a vector B four time the length of A pointing in the same direction as A. (c) Obtain a unit vector expression for a vector C three times the length of A pointing in the direction opposite the direction of A.
Solution: a) b) c)

11. [9] Three displacement vectors of a croquet ball are shown in Figure, where A = 20.0 units, B = 40.0 units, and C = 30.0 units. Find the magnitude and direction of the resultant displacement
Solution:

12. [10] Find the magnitude and the direction of resultant force
Solution: