Mechanics in 2 dimensions

1. Mechanics in 2 dimensions

2. Mechanics The branch of physics concerned with motion. • ‘kinematics’: describing motion • ‘dynamics’: explaining motion in terms of forces acting on particles.

3. Learning outcomes • analyse projectile motion in terms of a vertical acceleration plus constant horizontal speedand use this to solve quantitative problems • identify velocity & acceleration vectors at points on a projectile path • describe the conditions for static equilibrium when several forces act on a rigid object • analyse simple 2-D interactions using the law of conservation of momentum • draw and interpret free body force diagrams to represent forces acting on a particle or an extended rigid body • use Pythagoras’ theorem and trigonometry to resolve 2-D vectors into components and calculate resultants • use the main trigonometric functions of a calculator

4. Teaching challenges Students need careful guidance, discussion and practice to • understand what vectors and their components are (not only displacements, but also velocity, acceleration, angular displacement, angular velocity, force, field etc) • construct vector diagrams • understand functions sine, cosine and tangent • add and subtract vectors quantitatively using trigonometry

5. Warm-up In pairs: 1 Articulate clearly Newton’s three laws of motion. 2 For each law: Clarify when (how) it applies, with a few concrete examples. 3 Write down the kinematic equations, again clarifying when each of them applies. These BIG IDEASunderpin analysis of motion in 2D.

6. Trigonometric functions

7. Pythagoras

8. Adding perpendicular velocities A toy car with two motions: BIG IDEA: Velocities add as vectors. see VPL simulations Resolving vectors and Vectors

9. Adding vectors graphically Procedure 1. Place tail of 2nd vector at tip of 1st vector. 2. Vector sum is found as tail of 1st to tip of 2nd. (a parallelogram) Try the Geogebra simulation.

10. Adding vector components Try VPLab simulations Resolving vectors and Vectors

11. Projectile motion Object launched in a uniform gravitational field. BIG IDEA: Vertical & horizontal motions are independent. • vertically: uniform acceleration due to gravity • horizontally: constant speed

12. Parabolic path of a projectile Velocity changes by the same downward amount in each interval of time.

13. Parabolic path of a projectile

14. Resultant force • The need for a ‘free body force diagram’. Two tractors pull a barge along a straight canal. BIG IDEA: Forces add as vectors. The water will exert a drag force on the barge. What happens if the three forces are in equilibrium?

15. A closer look at the barge On a rigid body, a force can be applied with equal effect at any point along its line of action.

16. Block on an inclined plane

17. Distributed weight and resultant A ring. The resultant acts at its centre of mass C, which is in empty space!

18. Surface reaction & resultant Surface reaction is a distributed force, but it may be replaced for convenience by its resultant, Rn.

19. Momentum in 2 dimensions Linear momentum is conserved. BIG IDEA: Add momentum as vectors. see VPLab simulation 2D Collisions

20. Endpoints In pairs: Revisit the BIG IDEAS covered.