Simple Practical Mechanics. Lesley Swarbrick FMSP associate

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2 Simple Practical Mechanics Lesley Swarbrick FMSP associate

3 Practical Approaches to Mechanics

4 Mechanics is Applied Mathematics

5 Mechanics Experiments No Special Equipment

6 Kinematics

7 Resultant force: what direction? Extract from Mechanics in Action

8 Drop the Ball Drop a ball whilst walking fast and observe its horizontal motion. If you drop a ball out of the window of a moving car, where will it bounce?

9 Playing catch When the ball is in the air, what path does it follow? What forces are acting on the ball when it is in the air? How can you throw the ball so that it goes further?

10 Projectiles Mechanics in Action worksheet 28

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12 Forces and Newton s Laws

13 Short easy practicals Forces on stationary objects (N1 stationary, N3) Stand up and feel the forces on your feet. What forces are on you? Lift one foot off the floor. What happens to the forces? Push down on a table whilst stood up. What happens to the forces? Walk forward which way is friction acting?

14 Short easy practicals Net resultant force (N2) Two people push against each other. Who moves? As above but one on wheels/ice/socks on lino. Push a person. Then two people push the person which results in the same amount of force and in the same direction as the original push. This is a way of introducing the idea of components.

15 Newton s laws rephrased N1: No change in momentum No net force N2: A net resultant force in a direction change of momentum in a direction N3: Forces are subjective (I push down with 3N on you means you push up with 3N on me) N1 covers two situations: a stationary object and one with a constant velocity.

16 Short easy practicals How can you make a set of scales show less than your weight? How can you make a set of scales show more than your weight?

17 Bathroom scales and a broom Extract from Mechanics in Action

18 Bathroom scales and a broom Problem: Explain why the reading goes down when I press on the floor with a broom. Extract from Mechanics in Action

19 Draw a force diagram

20 W = Weight of the man Assumptions: Each body is a particle in equilibrium The scales and the brush have zero mass W

21 W Weight of the man F Force exerted on the brush F S N Contact force (brush and floor) Contact force (man and scales) F R Contact force (scales and floor) N R W N W S R S Hint: Think about pairs of forces (N3L)

22 Problem: Explain why the reading goes down when I press on the floor with a brush we need to show N is less than W

23 + F + N W = 0 F (N1L) Every particle continues in a state of rest or uniform motion in a straight line unless acted on by a resultant external force This means that for a body to be in equilibrium it must be the case that the resultant force acting on it is zero F + S F = 0 + R N = 0 R W N N W S R S

24 Interpreting these equations N + F W = 0 S F = 0 Therefore N = W F and F = S 0 so N W Assuming we are pushing with the broom N = W F means we can reduce N the harder we push down But N + F = W suggests F W so S W This means we can only push down with force W at most, as long as we are in equilibrium.

25 Bathroom scales and a broom Problem: If I press up on the ceiling, why does the reading go up? Extract from Mechanics in Action Problem: If I press down on the scales the reading is unchanged, why?

26 Sliding a graphical calculator on a desk Try to slide it button side up And button side down What do you notice? Can you explain? How could this be modelled mathematically?

27 The Friction Law F R

28 Friction Demonstrations Press the hands together gently. With a small reaction force the book can easily be pulled down as there is little friction. Press the hands together as hard as possible. With a large reaction force the book cannot easily be pulled down as there is a lot of friction.

29 Problem 1 What do you think a student or inexperienced teacher might say?

30 Problem 2 Pull a chair horizontally with a rope. Now pull the chair at an angle. Is it easier to pull the chair with the rope horizontal or at an angle?

31 Generally: Horizontally: T cos θ = F = R Vertically: T sin θ = Mg R Since on the point of sliding: F = R So T cos θ = Mg - T sin θ T (cos θ + sin θ) = Mg Because = tan λ where λ is the angle of friction: T = tan λ Mg/(cos θ + tan λ sin θ) T = sin λ Mg/(cos(λ θ)) For the least value of T, θ = λ

32 Moments

33 Rotating a door Stand by a door close to the hinge and see how much effort is required to open it. Now try pushing the door further away from the hinge, and note that the effort to open it is less. Now push further away from the hinge and see how much easier it gets. Recognise that this is a function of the distance from the hinge and the amount of force required to open the door. This also reinforces the idea that moments are turning forces about a pivot.

34 Sliding fingers together under a ruler What do you think will happen? Try it out were you right? Try starting your fingers at different positions on the ruler Can you explain? Can you suggest how to model this?

35 Balancing stacks of coins on a ruler Key concept: moments Stack coins to make the ruler balance.

36 Centres of Mass

37 Balancing

38 Balancing Fish (inspired by a project to design and build a fish mobile like this one )

39 First let s consider Balancing Whales

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41 x T M H t m h

42 x T M H t m h Area Distance T M H W t m h x Moments: Wx =Tt + Mm+ Hh Work out x

43 Now let s try Balancing Fish X Make a fish with a circular head and a triangular tail. THE FISH MUST BALANCE HORIZONTALLY WHEN SUSPENDED FROM THE POINT X

44 Finding the centre of mass of your fish tail hang it from a point and use a plumb line:

45 c Finding the centre of mass of your fish tail mathematically h c 1 3 h

46 I need to balance horizontally when suspended from point X. X? cm? cm.. but my head is TOO BIG!

47 For flat shapes, the weight is represented by the AREA. So the MOMENT is the product of area and distance of centre of mass from the pivot axis. Pivot axis Centre of Mass Centre of Mass Distance T Distance C Area T Area C For equal moments: Area T x Distance T = Area C x Distance C

48 Balance a Coke Can

49 Balance a Bottle

50 Newton Ball Tricks on YouTube

51 The experiments These experiments require little specialised apparatus. They are quick and easy to carry out. The questioning that follows the experiments is key to their effectiveness.

52 Discussion Why do practicals in mechanics? What is the value of short easy practicals? What do you think of the experiments in the session?

53 The Further Mathematics Support Programme Our aim is to increase the uptake of AS and A level Further Mathematics to ensure that more students reach their potential in mathematics. The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students. To find out more please visit