DHS 03 Dynamics (Lecture Notes & Tutorial)

Dunman High School (Senior High Physics) Topic 3 – Dynamics GUIDING QUESTIONS  How do forces affect the motion of an object?  When is momentum conserved during interactions between objects?  How can we analyse interactions using he principle of momentum conservation? Content • Newton’s laws of motion • Linear momentum and its conservation Learning Outcomes Students should be able to: (a) state and apply each of Newton's laws of motion (b) show an understanding that mass is the property of a body which resists change in motion (inertia) (c) describe and use the concept of weight as the force experienced by a mass in a gravitational field (d) define and use linear momentum as the product of mass and velocity (e) define and use impulse as the product of force and time of impact (f) relate resultant force to the rate of change of momentum (g) recall and solve problems using the relationship F = ma, appreciating that resultant force and acceleration are always in the same direction (h) state the principle of conservation of momentum (i) apply the principle of conservation of momentum to solve simple problems including inelastic and (perfectly) elastic interactions between two bodies in one dimension (knowledge of the concept of coefficient of restitution is not required) (j) show an understanding that, for a (perfectly) elastic collision between two bodies, the relative speed of approach is equal to the relative speed of separation (k) show an understanding that, whilst the momentum of a closed system is always conserved in interactions between bodies, some change in kinetic energy usually takes place.

2 Dunman High School Year 5 Physics 2024/2025 0. Introduction In kinematics, we described motion in terms of displacement, velocity a nd acceleration. In this chapter, we investigate what causes change in motion. The relationship between a force and the acceleration it causes was first understood by Sir Isaac Newton (1642 – 1727). The study of the relationship is called Newtonian mechanics. In this topic, the focus is on Newton’s three laws of motion. Newtonian mechanics does not apply in the following situations:  When the speeds of the interacting bodies are very large (i.e. approaching the speed of light). In t his case, Einstein’s special theory of relativity replaces Newtonian mechanics.  When the interacting bodies are on the atomic scale. In this case, quantum mechanics replaces Newtonian mechanics. Physicists now view Newtonian mechanics as a special case of these two more comprehensive theories. 1. Newton’s Laws of Motion LO (a) state and apply each of Newton's laws of motion LO (b) show an understanding that mass is the property of a body which res ists change in motion (inertia) Newton's 1st Law of Motion states that: It is also known as the Law of Inertia. Inertia is a measure of the reluctance of an object to change its state of rest or uniform motion in a straight line. When a force is applied on an object, it is always easier to change the state of motion of a lighter object (smaller mass). Likewise, it will be harder to stop a heavier object (bigger mass) that is moving. Newton’s First Law of Motion can also be used to explain the following observations: Observation Explanation A passenger in a moving bus will lurch forward when the bus driver suddenly brakes. When the bus suddenly brakes, the passenger tends to keep moving at the previous speed and lurch forward. A passenge r in a stationary bus will lurch backwards when bus driver suddenly accelerates. When the bus suddenly accelerates, the passenger tends to stay at rest and lurch backwards. An object at rest will remain at rest and an object in motion will remain in motion at constant velocity in the absence of an external resultant force. Mass is a measure of body’s inertia to changes in velocity. SI unit: kilogram (kg) Isaac Newton Hard to start moving Hard to stop moving once it is in motion

3 Dunman High School Year 5 Physics 2024/2025 LO (c) describe and use the concept of weight as the force experienced by a mass in the gravitational field weight = mass × acceleration of free fall W = mg Mass is a scalar quantity while weight is a vector quantity. Note that the mass of an object is constant all over the universe, but its weight is a force whose magnitude depends on the value of g. The direction of W is always in the direction of g. In the context of Earth, weight will always point towards the centre of the Earth. Example 1 Determine the acceleration of free fall on the surface of Earth and Moon if a 70.0 kg mass weighs 687 N on Earth and 114 N on Moon. weight / N acceleration of free fall near the planet’s surface / m s−2 Earth 687 gEarth = 687 / 70.0 = 9.81 Moon 114 gMoon = 114 / 70.0 = 1.63 LO (d) define and use linear momentum as the product of mass and velocity The linear momentum of a body is the product of the mass and its velocity. SI unit: kg m s-1 momentum = mass × velocity p = m v Momentum is a vector quantity. The direction of momentum is the same as the direction of the velocit y. (For the syllabus, the term momentum is used to mean linear momentum, unless specified.) Example 2 (a) Calculate the momentum of a 100 g bullet traveling at a speed of 400 m s−1 to the right. (b) Calculate the velocity required for a running person of mass 60.0 kg to have the same momentum as the bullet. Solution (a) momentum of bullet = (0.100)(400) = 40.0 kg m s-1 to the right (b) 40.0 = 60.0 v v = 0.667 m s-1 to the right The weight of a body is the force acting on the mass in a gravitational field. SI unit: newton (N)

4 Dunman High School Year 5 Physics 2024/2025 LO (f) relate resultant force to the rate of change of momentum Newton's 2nd Law of Motion states that: Mathematical Interpretation of Newton’s 2nd Law From Newton’s 2nd Law of Motion, ( ) R dp dp d mvF k k dt dt dt   where k is a constant of proportionality. For constant mass, then ( ) R d mv dvF k k m k madt dt     We make the constant k equals to 1 by defining the newton to be such that a resultant force of 1 N gives a mass of 1 kg an acceleration of 1 m s−2: RF ma (only valid for constant mass) R dpF dt (general form) Note that both FR and a are vectors – the force determines not only the magnitude of the acceleration, but also its direction – a body accelerates in the direction of the resultant force on it. Newton's 3rd Law of Motion states that: Newton’s 3rd Law of Motion implies that forces occur in pairs, which fulfils the following conditions: (a) The two forces act on two different bodies. (b) The forces are equal in magnitude. (c) The forces are opposite in direction. (d) The forces must be of the same type. These two forces are known as an action-reaction pair of forces. Examples: The rate of change of momentum of a body is directly proportional to the resultant force acting on the body and occurs in the direction of the resultant force. If body A exerts a force on body B, then body B exerts a force of the same type that is equal in magnitude and opposite in direction on body A.

5 Dunman High School Year 5 Physics 2024/2025 Example 3 In the free-body diagram of a book resting on a table, explain why the normal contact force, N, and the weight, W, do not form an action-reaction pair of forces even though N and W are equal in magnitude and opposite in direction. Example 4 A 0.50 kg ball experiences an acceleration of free fall of 9.81 m s −2. Find the acceleration of the Earth towards the ball (assume the ball and Earth are the only two objects in the universe).