HCI 2024 H3 Physics Complete Lecture Notes

Hwa Chong Institution (College) MOE H3 Physics 2024 1 Inertial Frames of Reference (non-relativistic) In the H3 syllabus, candidates should understand and apply concepts related to non-relativistic dynamics viewed from different inertial frames building on the understanding of collisions and the significance of the centre of mass in equilibrium situations. Learning Outcomes for the Inertial Frames (non-relativistic): Candidates should be able to: (a) show an understanding of what is meant by a frame of reference (b) recall and apply the Galilean transformation equations to solve problems relating observations in different frames of reference (c) show an understanding of what is meant by an inertial frame of reference, in the context of Newton’s laws of motion (d) show an understanding that the centre of mass moves as though the total mass is concentrated at that point and is acted upon by the net external force on the system (e) solve two-dimensional collision problems by considering velocities relative to the centre of mass of the system It will also be useful for this topic to recall the learning outcomes about the linear momentum and collisions in H2 syllabus. Students should be able to: (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 (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 The notes, figures and questions are the compilation from the following physics textbooks for the students who take H3 A level Physics 9814: • Young and Freedman, University Physics • Serway and Jerwett, Physics for Scientists and Engineers • Eric Mazur, Principles and Practices of Physics

Hwa Chong Institution (College) MOE H3 Physics 2024 2 Introduction Did you ever, sitting in a car at a red light and looking at the car next to you, slam on the brakes because you thought you were starting to roll but it turned out your car never moved? If you did, you experienced the relativity of motion, first described quantitatively by Galileo. The velocity measured for any object depends on the motion of the observer (the person doing the measuring). For example, to a person sitting in a moving train, a suitcase on the overhead rack is at rest, but to a person standing on a station platform watching the train speed by, the suitcase is not at rest. According to the person in the train, the suitcase has zero momentum and zero kinetic energy. According to the person on the platform, the suitcase has nonzero momentum and nonzero kinetic energy. In this topic, we investigate whether or not the laws of conservation of momentum and conservation of energy depend on the velocity of the observer. In other word s, if these laws are valid for one observer, are they also valid for an observer who is moving relative to the first observer? Frame of reference Whenever we talk about motion, we must specify a reference axis along which the motion occurs and an origin. A frame of reference is a combination of a reference axis that defines a direction in space and a reference point that defines the origin from which motion is measured. A frame of reference can be described by a Cartesian coordinate system for which an observer is at rest with respect to the origin. Let us conceptualize a sample situation in which there will be different observations for different observers. Consider the two observers A and B along the number line as in Fig. 1.1. Fig. 1.1 Observer A is located at the origin of a one dimensional x axis, while observer B is at the position 5Ax =− . Both observers measure the position of point P, which is located at 5Ax =+ .

Hwa Chong Institution (College) MOE H3 Physics 2024 3 Suppose observer B decides that he is located at the origin of an Bx axis as shown in Fig. 1.2. Fig. 1.2 Observer A claims point P is a located at a position with a value of +5, whereas observer B claims it is located at a position with a value of +10. Both observers are correct, even though they make different measurements. Their measurements differ because they are making the measurement from different frames of reference. Imagine now that observer B is moving to the right along x-axis. Now the two measurements are even more different. Observer A claims point P remains at rest a position with a value of +5, whereas observer B claims the position of P continuously changes with time, even passing him and moving behind him! Again, both observers are correct, with the difference in their measurements arising from their different frames of reference. Consider two observers watching a man walking on a moving beltway at an airport in Fig. 1.3. The woman standing on the moving beltway sees the man moving a normal walking speed. The woman observing from the stationary floor sees the man moving with a higher speed because the beltway speed combines with this walking speed. Both observers look at the same man and arrive at different values for his speed. Both are correct; the difference in their measurements results from the relative velocity of their frames of reference. You may watch the following video to have a better sense of about frames of reference. Fig. 1.3

Hwa Chong Institution (College) MOE H3 Physics 2024 4 Galilean transformation equations In a more general situation, consider a particle located at point P as shown in Fig. 1.4. Imagine that the motion of this particle is being described by two observers, observer A in a reference frame SA fixed relative to the Earth and a second observer B in a reference frame SB moving to the right relative to SA (and therefore relative to the Earth) with a constant velocity ABv . In this discussion of relative velocity, we use a double -subscript notation; the first subscript represents who is doing the observing, and the second represents w hat is being observed. Therefore, the notation ABv means the velocity of observer B (and the attached frame SB) as observed by observer A. With this notation, observer B measures A to be moving to the left with a velocity BA ABvv =− . Let us place each observer at his or her respective origin. Fig. 1.4 We define the time t = 0 as the instant at which the origins of the two reference frames coincide in space. Therefore, at time t, the origins of the reference frames will be separated by a distance ABvt . We label the position P of the particle relative to observer A with the position vector APr and that relative to observer B with position vector BPr , both at time t. From Fig. 1.4, we see that the vectors APr and BPr are related to each other through the expression AP BP ABr r v t=+ (1.1) By differentiating the Equation (1.1) with respect to time, noting that v is constant, we obtain AP BP AB AP BP AB dr dr vdt dt u u v =+ =+ (1.2) where APu is the velocity of the particle at P measured by observer A and BPu is its velocity measured by B. Equations (1.1) and (1.2) are known as Galilean transformation equations. They relate the position and velocity of a particle as measured by observers in relative motion. The Galilean

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