AS Leveel Dynamics 9702 Physics syllabus

AS Dynamics

1. Representing forces as vectors  A force is a push or a pull measured in Newtons. One force we are very familiar with is the force of gravity , AKA the weight . The very concepts of push and pull imply direction. Thus forces are vectors .  The direction of the weight is down toward the center of the earth. If you have a weight of 90 Newtons (or 90 N), your weight can be expressed as a vector: 90 N, down. We will show later that weight has the formula AS Dynamics Forces W = mg weight where g = 10 m s -2 and m is the mass in kg

2. Free-body diagram Objects as point particles and Free-body diagrams AS Dynamics Forces Kindly note in exam use g = 9.81 m/s 2 EXAMPLE: Calculate the weight of a 25-kg object. SOLUTION: Since m = 25 kg and g = 10 m s -2 , W = mg = (25)(10) = 250 N (or 250 n). Note that W inherits its direction from the fact that g points downward. We sketch the mass as a point particle (dot), and the weight as a vector in a free-body diagram : mass force W W = mg weight where g = 10 m s -2 and m is the mass in kg

3. Objects as point particles and Free-body diagrams AS Dynamics Forces  Certainly there are other forces besides weight that you are familiar with. For example, when you set a mass on a tabletop, even though it stops moving, it still has a weight.  The implication is that the tabletop applies a counterforce to the weight, called a normal force . Note that the weight and the normal forces are the same length – they balance. The normal force is called a surface contact force. W R FYI  The normal force is often called (unwisely) the reaction force – thus the R designation.

4. Objects as point particles and Free-body diagrams  Tension T can only be a pull and never a push.  Friction F f tries to oppose the motion.  Friction F f is parallel to the contact surface .  Normal R is perpendicular to the contact surface .  Friction and normal are mutually perpendicular . F f   R .  Friction and normal are surface contact forces .  Weight W is an action-at-a-distance force . AS Dynamics Forces T the tension W R F f Contact surface

5. Other forces to know about:  Drag Upthrust AS Dynamics Forces

6. Sketching and interpreting free-body diagrams  Weight is sketched from the center of an object. Normal is always sketched perpendicular to the contact surface. Friction is sketched parallel to the contact surface. Tension is sketched at whatever angle is given. AS Dynamics Forces T W R F f

7. EXAMPLE: An object has a tension acting on it at 30° as shown. Sketch in the forces, and draw a free-body diagram. SOLUTION: Weight is drawn from the center, down. Normal is drawn perpendicular to the surface from the surface. Friction is drawn par- allel to the surface. Free-body diagram Sketching and interpreting free-body diagrams AS Dynamics Forces T 30 ° W R F f T 30 ° W R F f

8. Solving problems involving forces and resultant force The resultant (or net) force is just the vector sum of all of the forces acting on a body. AS Dynamics Forces EXAMPLE: An object has mass of 25 kg. A tension of 50 n and a friction force of 30 n are acting on it as shown. What is the resultant force? SOLUTION: Since the weight and the normal forces cancel out in the y-direction, we only need to worry about the forces in the x-direction. The net force is thus 50 – 30 = 20 n (+x-dir). T W R F f 50 n 30 n

9. Unbalanced & Balanced force s The resultant (or net) force is just the vector sum of all of the forces acting on a body. AS Dynamics Forces Forces on an object are balanced when the resultant force on the object is zero. The object will either remain at rest or have a constant velocity.

10. AS Dynamics Forces Moving through fluids In general, whenever a body moves through a fluid (gas or liquid) it experiences a fluid resistance force that is directed opposite to the velocity. Typically, F = kv for low speeds and F = kv 2 for high speeds (where k is a constant). The magnitude of this force increases with increasing speed.

11. Describing the consequences of Newton’s first law for translational equilibrium  Newton’s first law is drawn from his concept of net force and Galileo’s concept of inertia. Newton’s first law says that the velocity of an object will not change if there is no net force acting on it. In his words... “Every body continues in its state of rest , or of uniform motion in a straight line , unless it is compelled to change that state by forces impressed thereon.” In symbols...   F = 0 is the condition for translational equilibrium . AS Dynamics Forces v = 0 v = CONST  F If  F = 0, Newton’s first law then v = CONST. A body’s velocity will only change if there is a net force acting on it.

12. Translational equilibrium  As a memorable demonstration of inertia – matter’s tendency to not change its state of motion (or its state of rest) - consider this: A water balloon is cut very rapidly with a knife. For an instant the water remains at rest! AS Dynamics Forces

13. Solving problems involving forces and resultant force AS Dynamics Forces EXAMPLE: A 1000-kg airplane is flying at a constant velocity of 125 m s -1 . Label and determine the value of the weight W , the lift L , the drag D and the thrust F if the drag is 25000 N. SOLUTION: Since the velocity is constant, Newton’s first law applies. Thus  F x = 0 and  F y = 0. W = mg = 1000(10) = 10000 N ( down ).  Since  F y = 0, L - W = 0, so L = W = 10000 N ( up ) .  D = 25000 N tries to impede the aircraft ( left ).  Since  F x = 0, F - D = 0, so F = D = 25000 N ( right ) . W L D F

14. If mass is measured in kg and velocity in m s –1 , what are the units of momentum? p = m v momentum = mass × velocity The units are kg m s –1 . (This can also be expressed as N s ). Momentum is a property of objects with mass and velocity. It is a vector quantity with the same direction as the velocity of the object. What is momentum?

15. AS Dynamics Forces In a tennis match, when a player exerts a force on the ball, it changes momentum. This means the ball can change speed, direction, shape or size, etc. In order to change the momentum of an object, a force must be applied (from Newton’s first law). The rate of change of momentum of an object is proportional to the resultant force acting on the object. The resultant force and the change in momentum are in the same direction. This is an alternative way of stating Newton’s second law in terms of momentum.

16. AS Dynamics Forces Stated mathematically, Newton’s second law is: For constant masses this becomes, F = m Δ v Δ t = m a It can therefore be seen that the familiar equation F = m a is a special case of the more general equation for Newton’s second law in terms of momentum. The more general form of the equation is necessary when mass is not constant, for example for a space shuttle taking off. The mass decreases as fuel is burned. F = Δ p Δ t

17. AS Dynamics Forces Calculate the average force acting on a 900 kg car when its velocity changes from 5.0 m s −1 to 30 m s −1 in a time of 12 s.

18. AS Dynamics Forces Newton’s laws of motion – The second law F net = ma Newton’s second law (or  F = ma ) PRACTICE: Use F = ma to show that the formula for weight is correct. SOLUTION: F = ma . But F is the weight W. And a is the freefall acceleration g . Thus F = ma becomes W = mg .

19. Newton’s laws of motion – The second law F net = ma Newton’s second law (or  F = ma ) AS Dynamics Forces EXAMPLE: A 1000-kg airplane is flying in perfectly level flight. The drag D is 25000 n and the thrust F is 40000 n. Find its acceleration. SOLUTION: Since the flight is level,  F y = 0.   F x = F – D = 40000 – 25000 = 15000 n = F net . From F net = ma we get 15000 = 1000 a , or a = 15000 / 1000 = 15 m s -2 . W L D F

20. A skydiver falling freely https://www.youtube.com/watch?v=EabUUrZFnFE AS Dynamics Forces At the start of the fall, the only force acting on the diver is his or her weight. The acceleration of the diver at the start must therefore be g . Then increasing air resistance opposes their fall and their acceleration decreases. Eventually, they reach terminal velocity . At the terminal velocity, the air resistance is equal to the weight. The terminal velocity is approximately 50 m s -1 , but it depends on the skydiver’s weight and orientation. Head-first is fastest.

21. Newton’s laws of motion – The third law In words “For every action force there is an equal and opposite reaction force.” In symbols In the big picture, if every force in the universe has a reaction force that is equal and opposite, the sum of all the forces in the whole universe is zero! AS Dynamics Forces F AB = - F BA F AB is the force on body A by body B. F BA is the force on body B by body A. Newton’s third law FYI So why are there accelerations all around us?  Because each force of the action-reaction pair acts on a different mass .

22. Newton’s laws of motion – The third law In words “For every action force there is an equal and opposite reaction force.”  AS Dynamics Forces

23. Each body acts in response only to the force acting on it . The door CAN’T resist F A B , but you CAN resist F B A . Identifying force pairs in context of Newton’s third law AS Dynamics Forces EXAMPLE: When you push on a door with 10 n, the door pushes you back with exactly the same 10 n, but in the opposite direction. Why does the door move, and not you? SOLUTION: Even though the forces are equal and opposite, they are acting on different bodies. A B F A B A F B A the door’s reaction your action

24. If mass is measured in kg and velocity in m s –1 , what are the units of momentum? p = m v momentum = mass × velocity The units are kg m s –1 . (This can also be expressed as N s ). Momentum is a property of objects with mass and velocity. It is a vector quantity with the same direction as the velocity of the object. What is momentum?

25. Since momentum is the product of mass and velocity, an object ' s momentum changes whenever its mass or velocity changes. p is sometimes referred to as the linear momentum to distinguish it from angular momentum, a quantity associated with a rotating object. Momentum is a vector quantity. The momentum vector points in the same direction as the velocity vector. AS Dynamics Momentum

26. The figure below shows two objects, a beanbag bear and a rubber ball, each with the same mass and same downward speed just before hitting the floor. What is the change in momentum of each of the objects? AS Dynamics Momentum

27. If the beanbag has a mass of 1 kg and is moving downward with a speed of 4 m/s just before coming to rest on the floor, then its change in momentum is A 1-kg rubber ball with a speed of 4 m/s just before hitting the floor will bounce upward with the same speed. Therefore, the ball ' s change in momentum is AS Dynamics Momentum

28. AS Dynamics Conservation of linear momentum The principle of conservation of linear momentum states: The total linear momentum of a system of interacting bodies is constant, providing no external forces act. This applies to collisions , where objects move together and hit one other, and to explosions , where objects fly apart from one another after initially being at rest. collision explosion

29. For a closed system where no resultant external force acts , in any direction: T otal momentum of objects before collision = Total momentum of objects after collision AS Dynamics Conservation of linear momentum The principle of conservation of linear momentum can also be stated as:

30. The figure below shows both the internal and external forces acting on a rider and bicycle. Internal forces, such as a push on the handlebars exerted by a bicycle rider, act between objects within a system. External forces, such as the force the road exerts on a rear bicycle tire, are exerted on the system by something outside the system. AS Dynamics Conservation of linear momentum

31. In Figure, trolley A of mass 0.80 kg travelling at a velocity of 3.0 m s −1 collides head-on with a stationary trolley B. Trolley B has twice the mass of trolley A. The trolleys stick together and have a common velocity of 1.0 m s −1 after the collision. Show that momentum is conserved in this collision. AS Dynamics Conservation of linear momentum

32. A collision occurs when two objects free from external forces strike one another. Examples of collisions include one billiard ball hitting another, a baseball bat hitting a ball, and one car smashing into another. Momentum is conserved when objects collide. However, this does not necessarily mean that kinetic energy is conserved as well. Collisions are categorized according to what happens to the kinetic energy of the system. Collision lab https://phet.colorado.edu/sims/html/collision lab/latest/collision-lab_en.html AS Dynamics Collision

33. A collision in which the kinetic energy is conserved is referred to as an elastic collision. In an elastic collision, the final kinetic energy of the system is equal to its initial kinetic energy. A collision in which the kinetic energy is not conserved is called an inelastic collision. In an inelastic collision, the final kinetic energy is less than the initial kinetic energy. An inelastic collision where the colliding objects stick together is referred to as a completely inelastic collision. AS Dynamics Collision

34. Most everyday collisions are far from elastic. However, objects that bounce off each other with little deformation—like billiard balls—provide a good approximation to an elastic collision. AS Dynamics Collision When one ball is swung on Newton’s cradle , one ball moves out at the other end. If two balls are swung, two balls move out.

35. Springy collisions P erfectly elastic collision AS Dynamics Collision In a perfectly elastic collision of two bodies, the relative speed of the body’s approach is equal to the relative speed of their separation. In a perfectly elastic collision of two bodies, the relative speed of the body’s approach is equal to the relative speed of their separation.

36. Sticky collisions I nelastic collision AS Dynamics Collision During an inelastic collision, the total kinetic energy of the bodies becomes smaller. In inelastic collisions, momentum is conserved. However, kinetic energy is not conserved. It is lost because work is done in deforming the two objects.

37. AS Dynamics Collision

38. AS Dynamics Collision

39. Momentum conservation applies to the largest possible system—the universe. The exploding star in the photo below sends material out in opposite directions, thus ensuring that its total momentum is unchanged. AS Dynamics Momentum

40. Momentum conservation may cause objects to recoil. Recoil is the backward motion caused by two objects pushing off one another. Recoil occurs when a gun is fired or, as is shown in the figure below, when a firefighter directs a stream of water from a fire hose. In all cases, recoil is a result of momentum conservation. AS Dynamics Momentum

41. If you push a large rock over a cliff, its speed increases as it falls. Where does its momentum come from? And when it lands, where does its momentum disappear to? The rock falls because of the pull of the Earth’s gravity on it. This force is its weight, and it makes the rock accelerate towards the Earth. Its weight does work and the rock gains kinetic energy. It gains momentum downwards. The Earth starts to move upwards as the rock falls downwards. The mass of the Earth is so great that its change in velocity – far too small to be noticeable. AS Dynamics Momentum

42. If a rock of mass 60 kg is falling towards the Earth at a speed of 20 m s -1 , how fast is the Earth moving towards it? The mass of the Earth is 6.0 × 10 24 kg. The minus sign shows that the Earth’s velocity is in the opposite direction to that of the rock. The Earth moves very slowly indeed AS Dynamics Momentum

43. AS Dynamics Collisions in two dimensions At first, the white ball is moving straight forwards. When it hits the red ball, it moves off to the right and i ts speed decreases. The red ball moves off to the left. It moves off at a bigger angle than the white ball, but more slowly. Momentum is a vector quantity and so we can split it into components

44. A white ball of mass m = 1.0 kg and moving with initial speed u = 0.5 m s -1 collides with a stationary red ball of the same mass. They move off so that each has the same speed and the angle between their paths is 90°. What is their speed? AS Dynamics Collisions in two dimensions

45. AS Dynamics Collisions in two dimensions

46. Figure below shows the momentum vectors for particles 1 and 2, before and after a collision. Show that momentum is conserved in this collision. AS Dynamics Collisions in two dimensions

47. AS Dynamics Collisions in two dimensions

48. AS Dynamics Collisions in two dimensions