1. Force as a vector quantity
Mechanics Chapter 10
Force as a vector quantity

2. 10.1 Combining forces geometrically
Vectors – have both magnitude and directions, i.e. displacement, velocity, acceleration and force. Vectors are symbolized with bold face. Scalars – only magnitude, no direction is associated. i.e. mass, energy.

3. 10.1 Combining forces geometrically
Adding Vectors and finding resultant vectors

4. 10.1 Combining forces geometrically
Triangle Law for combing Forces - If two forces P and Q are represented by arrows and the arrow representing R is obtained from these, then the single force R has exactly the same effect on a particle as the two forces P and Q acting together.

5. 10.1 Combining forces geometrically
Find the resultant R of the two forces P and Q by calculation.

6. 10.1 Combining forces geometrically
The resultant of two forces with magnitudes P and Q in perpendicular directions is a force of magnitude 𝑃 2 + 𝑄 2 which makes and angle tan −1 𝑄 𝑃 with the direction of the force P.

7. 10.2 Splitting a force into components
Khalib and Trey are pushing a piano across a stage. This needs a force of 240 N. Trey pushes at 20 degrees to the left of the desired direction of motion, Khalib pushes at 30 degrees to the right of it. How hard must each push to move the piano at these angles?

8. 10.2 Splitting a force into components
A ship is towed along a narrow channel by cables attached to two tugboats. The more powerful tugboat produces a force of 800 kN; its cable is at 10 degrees to the direction of the channel. The other tugboat is to produce as small a force as possible. What should be the direction of the second tugboat’s cable, and how large is the net forward force on the ship?

9. 10.2 Splitting a force into components
If a force is split into components in two perpendicular directions, the magnitude of each component is the resolved part of the force in that direction.

10. 10.3 Combining forces by perpendicular components
Step 1: Choose two directions at right angles Step 2: Split each force into components in these directions Step 3: For each direction, find the sum of the components you have calculated Step 4: Find the resultant of the two sums

11. 10.3 Combining forces by perpendicular components
Calculate the resultant of P and Q

12. 10.3 Combining forces by perpendicular components
The following four forces act on a particle. Find their resultant V. P – 6 N in the x-direction Q – 8 N at 50 degrees to the x-direction S – 12 N in the negative y-direction U – 10 N at 160 degrees to the x-direction

13. 10.5 Equilibrium
Triangle of Forces – If three forces, X, Y and Z are in equilibrium, then the corresponding vectors can be represented by the sides of a closed triangle in which at each vertex the head of one vector arrow meets the tail of another.

14. 10.5 Equilibrium
Triangle of Forces – If three forces, X, Y and Z are in equilibrium, then the corresponding vectors can be represented by the sides of a closed triangle in which at each vertex the head of one vector arrow meets the tail of another.

15. 10.5 Equilibrium
Three strings are knotted together at one end, and parcels of weights 5N, 7N, and 9N are attached to the other ends. The first two strings are placed over smooth horizontal pegs, and the third parcel hangs freely. The system is in equilibrium. Find the angles which the first two strings make with the vertical between the knot and the peg.

16. 10.5 Equilibrium
A trough is formed from two rectangular planks at angles of 30 and 40 degrees to the horizontal, joined along a horizontal edge. A cylindrical log of weight 400 N is placed in the trough with its axis horizontal. Calculate the magnitudes P and Q of the normal contact forces from each plank on the log
Using a triangle of forces
By resolving

17. 10.5 Equilibrium
If two forces of unknown magnitude act on a particle in equilibrium, then can be found directly by resolving in directions perpendicular to each force.

18. 10.5 Equilibrium
A coordinate grid is marked on a vertical wall. Small smooth pegs are driven into the wall at the points (-3, 11) and (9, 7) and a hoop of weight W and radius rests on the pegs with its center at the origin. Find the magnitudes of the contact forces at the pegs in terms of W.