ASRJC Dynamics Notes

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 / 8867 1 Additional Notes Topic 3: Dynamics Content: • Newton’s laws of motion • Linear momentum and its conservation Learning Outcomes: Candidates should be able to: Newton’s laws of motion (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 effect of a gravitational field on a mass. Linear momentum and its conservation (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 the 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 system is always conserved in interactions between bodies, some change in kinetic energy usually takes place.

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 / 8867 2 Additional Notes Introduction • In Kinematics, we described motion in terms of position, velocity and acceleration without considering what might cause that motion. • In Dynamics, we consider the cause – what might cause one object to remain at rest and another object to accelerate? The two main factors are the mass of the body, and the forces acting on the body. Dynamics is therefore a study of the relationship between the forces acting on a body and the motion of that body. • The study of dynamics is largely governed by Newton’s Three Laws of Motion. A.1 Newton’s 1st Law • Newton’s 1st Law is also known as the law of inertia. • The inertia of a body is the property of a body to resist a change in its motion. • The mass of a body is a measure of the resistance of a body to a change in its motion i.e. a measure of its inertia. • The greater the mass of a body, the lesser the body accelerates under the action of a given applied force. • Unlike weight (see side note) which depends on gravitational field and therefore varies from place to place, mass is independent of gravitational forces, i.e. the mass of a body is the same on the Earth as on Moon. Mass is a scalar quantity while weight is a vector. Newton’s First Law (Law of inertia): A body stays at rest or continues to move with constant velocity unless a resultant force acts on it. Newton’s Second Law: The rate of change of momentum of a body is proportional to the resultant force acting on the body and takes place in the direction of the resultant force. Newton’s Third Law: If body A exerts a force on body B, then body B will exert a force of the same type that is equal in magnitude and opposite in direction on body A. Nature of Science Science is an evidence-based, model-building enterprise concerned with the natural world. In this topic, we can see how models such as free-body diagrams and vector diagrams are used to represent real-life scenarios. Using models allow us to make predictions, which we can proceed to verify experimentally. Relating Science and Society Examples of application of Newton’s Law of Motion are found everywhere in everyday life. These can range from simple ones such as walking, playing tennis, to more complex applications such as car safety and rocket building. A body stays at rest or continues to move with constant velocity unless a resultant force acts on it. memorise The weight W of a body is the gravitational force exerted on the body. If g is the acceleration of the body towards the centre of the Earth then we can substitute F (force accelerating the body) = W and a = g in F = ma, hence W = mg For N2L defn, it is wrong to interchange the terms “rate of change of momentum” with “resultant force”.

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 / 8867 3 Additional Notes Worked Example 1 A passenger in a bus claimed that a bag flew forward when the bus brakes. Explain how this is possible. A.2 Newton’s 2nd Law A.2.1 Linear Momentum To understand Newton’s 2nd Law, first, we need to know what linear momentum is. Mathematically, it is p = mv where p is the linear momentum, m is the mass, and v is the velocity of the body. • Linear momentum is a vector quantity. • Its direction is along its velocity v and its SI unit is kg m s−1 or N s. • Momentum is a quantity that describes objects in motion. Imagine that you have intercepted a football and see two players A (of mass 80 kg) and B (of mass 130 kg) from the opposing team approaching you as you run with the ball. Both of the players are running towards you at 5 m s−1. However, because the two players have different masses, intuitively you know that you rather collide with player A rather than player B. A.2.2 Newton’s 2nd Law • Mathematically, this translates to: F  t mumv − where v is the final velocity, and u is the initial velocity. F  m a if a is the acceleration of the body, then a = (v − u) / t F = k ma where k is a constant which has no unit but whose size depends on the unit to be chosen for force. • According to N1L, since no horizontal force acts on the bag during braking, it will continue to move forward with the same velocity as before. • Since the bus slows down as it brakes while the bag maintains its horizontal velocity, the bag is seen by the passenger as flying forward. Linear momentum is defined as the product of the mass of the body and its velocity. The rate of change of momentum of a body is proportional to the resultant force acting on the body and takes place in the direction of the resultant force. memorise memorise Do note that Newton’s 2nd Law is not defined as F = ma! That was good enough for O- Level but not good enough for A-Level.

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 / 8867 4 Additional Notes • With SI units, the newton (N) is the unit of force. One newton is defined as the force which gives a mass of 1 kg an acceleration of 1 m s–2. Substituting F = 1 N, m = 1 kg and a = 1 m s–2 in F= k ma we obtain k = 1. Thus with these units k = 1 and F = ma. • Thus, a force can be measured by finding the acceleration it produces in a known mass. • If SI units are used, Newton’s second law may be written as : Resultant force = rate of change of momentum net dp d(mv)F = =dt dt • If the force is applied over a time interval of Δt : Average resultant Force,   net pF= t Worked Example 2 A body of mass 6.0 kg initially has a velocity 4.0 m s–1 in the eastward direction. It suddenly changes velocity to 3.0 m s–1 in the southward direction in a time of 0.20 s. Calculate the average force acting on the body during the change of direction. Remember all the change in velocity Δv questions from Topic 1 Measurements? Remember that Δv is a vector. Here is where you put your ability to do vector subtraction to good use. Change in velocity, Δv = final velocity vf – initial velocity vi Δv = vf + (–vi) Δv − 22 1 = 3.0 + 4.0 = 5.0 m s 4.0tan 3.0θ= =>  = 53.1°. By N2L, Average Force, p m vF= tt  