Physics 2 Second Quarter Reviewer PDF

Physics 2: Second Quarter Reviewer Vito “V” Ravago Start Date: 11/10/2024 End Date: ?? Table of Contents: 1. Pendulum Definition of Terms Period and its Derivation 2. Waves Properties of Waves.l..LL.. NOTE 1: Most o...

Physics 2: Second Quarter Reviewer Vito “V” Ravago Start Date: 11/10/2024 End Date: ?? Table of Contents: 1. Pendulum Definition of Terms Period and its Derivation 2. Waves Properties of Waves.l..LL.. NOTE 1: Most of this stuff comes from Giancoli’s Physics Principles & Applications Pendulum A simple pendulum consists of a small object (or a bob) suspended from a lightweight cord or rod. In a simple pendulum, we assume that the cord does not stretch and that its mass (of the rod) can be neglected. We also assume that the pivot is frictionless. In a closed system (or isolated system), the motion of the pendulum resembles simple harmonic motion (SHM). FIGURE 1.1: Parts of a Pendulum In a pendulum, if the massive bob is suspended with an angular displacement of Θ, it will reach the other side also with an angular displacement of Θ. (remember the law of conservation of energy). We treat this point as the point of origin. If the angular displacement or Θ is equal to 0 radians, the bob is in the equilibrium position, and has maximum kinetic energy. (or when the bob is at its lowest position).  FIGURE 1.2: Energy Level Depictions of a Pendulum On the highest points of the bob’s trajectory (point A and G), the bob has 0 kinetic energy and maximum potential energy. At point D or the equilibrium position, the bob undergoes maximum velocity, meaning that all of its energy is kinetic. (also 0 potential energy) As the pendulum oscillates from Point A, the potential energy will convert into kinetic energy, and at the equilibrium point, the kinetic energy will convert into potential energy (from point D to G). The formulas for potential kinetic energy and kinetic energy of a pendulum in SHM are: 𝑃. 𝐸 = 𝑚𝑔ℎ 1 2 𝐾. 𝐸 = 2 𝑚𝑣 𝑚 = 𝑚𝑎𝑠𝑠 𝑔 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 ℎ = ℎ𝑒𝑖𝑔ℎ𝑡 𝑣 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑏  FIGURE 1.3: Free Body Diagram of a Pendulum In 1.3, the variables are: 𝐿 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑜𝑑 (𝑚) Θ = 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 (𝑟𝑎𝑑𝑖𝑎𝑛𝑠) 𝑚 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑏 𝑔 = 𝑎𝑐𝑐. 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 s 𝑠 = 𝑙𝑖𝑛𝑒𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑎𝑙𝑜𝑛𝑔 𝑡ℎ𝑒 𝑎𝑟𝑐 𝑜𝑟 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑟𝑐 The linear displacement along the arc is given by the formula: 𝑠 = 𝐿Θ (equation 1) take note that π 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = ( 180 )(𝑛𝑜. 𝑜𝑓 𝑑𝑒𝑔𝑟𝑒𝑒𝑠) If you pay attention to the graph, we can see that the force of tension (𝐹𝑇) and the y-component (𝑚𝑔𝑐𝑜𝑠Θ), cancel each other out. This means that the ball doesn’t accelerate along the length of the string. If the restoring force is proportional to the length of the arc or the angular displacement, the motion will be simple harmonic.  With this knowledge, we can assume that the restoring force is equal to the component of weight tangent to the arc (or s), which is 𝑚𝑔𝑠𝑖𝑛Θ. The formula of the restoring force is: 𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 = − 𝑚𝑔𝑠𝑖𝑛Θ remember that restoring force is always negative, going towards the direction opposite the angular displacement. always remember that theta is expressed in RADIANS. If the angular displacement is small enough (Θ < 0. 349 𝑟𝑎𝑑) the sine of Θ and Θ is very nearly equal to each other when specified in radians. This can be seen in Figure 1.3, where the arc length (𝑠 = 𝐿θ) is nearly the same length as the cord (s ≈ LsinΘ). We can approximate the restoring force as the formula below (for angles lesser than 0.349 rads): 𝐹 ≈ − 𝑚𝑔Θ (equation 2) 𝑠 substituting Θ for 𝐿 (which comes from 𝑠 = 𝐿Θ (equation 1)), we have 𝑚𝑔 𝐹≈ − 𝐿 𝑠 (equation 3) To derive for period, the motion of a pendulum can be approximated as simple harmonic, since the approximate equations listed above fits the model of Hooke’s Law, 𝑚𝑔 which is 𝐹 = − 𝑘𝑥. Instead of x, we have arc length s, and instead of k, we have 𝐿. (coming from equation 3). The formula below is the period of a pendulum:  𝑚 𝑇 = 2π 𝑘 substitute k 𝑚 𝑇 = 2π 𝑚𝑔 𝐿 𝐿 𝑇 = 2π 𝑔 since time and frequency are inversely proportional to each other, frequency is: 1 𝑔 𝑓 = 2π 𝐿 Waves Waves are disturbances that transfer energy without any net matter displacement. There are two types of waves: Mechanical and Non-Mechanical. 1. Mechanical Waves: are waves that require a medium to transfer energy. Examples of these kinds of waves are: Longitudinal waves, transverse waves, and surface waves. 2. Non-Mechanical Waves: are waves that do not require a medium to transfer energy. They travel through vacuums. Examples of these kinds of waves are: Electromagnetic waves, gravitational waves. Waves are formed with a pulse. For example, a single pulse can be formed on a cord with a quick up-and-down motion of the hand. Because the end section (or the other end of the cord) is attached to adjacent sections, the cord will also feel an upward force, and they will also have a kind of upward or downward force (depending on what direction you whip the cord). (remember Newton’s second law of motion). As each succeeding section of the cord moves upward, a wave crest moves outward along the cord. As each crest reaches peak position or maximum displacement, they will return to the equilibrium position due to tension from the adjacent section of the cord, and a disturbance (or a wave) is made which makes the pulse travel.  FIGURE 2.1: A wave traveling on a rope. The particle then oscillates back and forth on the table top as shown. To clear out, the velocities of a wave and the rope particle as shown above are not equal. The wave of the rope moves to the right along the tabletop, but each piece of the rope (particle) only vibrates to-and-fro, perpendicular to the traveling wave. Thus, the rope is not moving in the same direction as the wave. Although a wave does not matter itself, it can travel through. Therefore, a wave consists of oscillations that move without carrying matter. Properties of Waves: There are important quantities that define a rope, namely: 1. Amplitude (A): it is the maximum height of a wave crest, or maximum depth of a trough. There are two types of amplitudes, the crest (high points), and the troughs (the low points). 2. Wavelength (λ): it is the distance between two successive crests. It is also equal to the distance between any two successive identical points on a wave. 3. Period (T): the time elapsed between two successive crests passing by the same point in space. It is inversely proportional to frequency. 4. Frequency (f): the number of cycles that pass through a given point per unit time. 5. Wave Speed (v): the speed at which wave crests (or any other type of fixed point on the wave shape) move forward. A wave crest travels a distance of one wavelength (λ) in a time equal to one period (T). Thus the formula/s for wave velocity is: 𝑣 = λ𝑓 1 𝑣 = λ 𝑇 1 𝑔 𝑣 = (λ)( 2π 𝐿 ) 𝐹𝑇𝑒𝑛𝑠𝑖𝑜𝑛 𝑣 = µ𝑀𝑎𝑠𝑠 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝐹𝑇𝑒𝑛𝑠𝑖𝑜𝑛 𝑣 = 𝑚𝑎𝑠𝑠 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑅𝑜𝑝𝑒 𝑚𝑎𝑠𝑠 𝑑𝑜𝑛'𝑡 𝑓𝑜𝑟𝑔𝑒𝑡 𝑡ℎ𝑎𝑡 µ𝑀𝑎𝑠𝑠 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 = 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑅𝑜𝑝𝑒 6. Intensity of a Wave (I): is the time-average rate at which energy is transported by a wave per unit area, across a surface perpendicular to the direction of 𝑊 𝐽 propagation. The S.I unit for intensity is 2 𝑜𝑟 2 𝑚 𝑚 *𝑠 FIGURE 2.2: Wave Propagation or Wave Movement  Recall: 1 2 𝑃𝐸𝑚𝑎𝑥 = 2 𝑘𝐴 2 we can say that energy is DIRECTLY PROPORTIONAL to 𝐴 (position). This can be seen in Figure 2.2 above. 2 𝐸∝𝐴 Recall: 𝐸 𝑃 = 𝑡 Power is equal to Energy over Time. If we replace E with the formula for maximum potential energy, we get 1 2 1 𝑃 = 2 𝑘𝐴 * 𝑡 2 thus we can also say that power is DIRECTLY PROPORTIONAL to position or 𝐴 Given the properties and definition of intensity, the formula for intensity is: 𝑃 𝐼 = 𝑎 𝐸 𝑟𝑒𝑚𝑒𝑚𝑏𝑒𝑟, 𝑃 = 𝑡 𝑃 = 𝑝𝑜𝑤𝑒𝑟 𝑎 = 𝑎𝑟𝑒𝑎 𝐼 = 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 𝐸 = 𝑒𝑛𝑒𝑟𝑔𝑦 𝑡 = 𝑡𝑖𝑚𝑒 Again, we can substitute power for equation 2.1, so we can also say that intensity is DIRECTLY PROPORTIONAL to the position. 2 𝐼∝𝐴  FIGURE 2.3: Spherical Wave If a wave flows out in all directions, a spherical wave forms, so the energy spreads out over a larger area. Figure 2.3 is an example of this. Since the area or a of a 2 sphere is equal to 4π𝑟 , so: 𝑃 𝐼𝑠𝑝ℎ𝑒𝑟𝑒 = 2 4π𝑟 2 so we can say that intensity is inversely proportional to 𝑟 or radius since the surface 2 area of the sphere is directly proportional to 𝑟. DEEPER ANALYSIS: If the power output P of the source is CONSTANT, then the intensity decreases as the inverse square of the distance from the source. Inverse-Square Law for Proportionality: If no energy is absorbed between the two spheres, the power must be equal for both. As distance increases, intensity decreases, and vice versa. 2 𝐼1 𝑟2 𝐼2 = 2 𝑟1 𝐼1 = 𝐼𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 𝑎𝑡 𝑃𝑜𝑖𝑛𝑡 1 𝐼2 = 𝐼𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 𝑎𝑡 𝑃𝑜𝑖𝑛𝑡 2 𝑟1 = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑠𝑜𝑢𝑟𝑐𝑒 𝑡𝑜 𝑃𝑜𝑖𝑛𝑡 1 𝐼1 = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑠𝑜𝑢𝑟𝑐𝑒 𝑡𝑜 𝑃𝑜𝑖𝑛𝑡 2 The amplitude of a wave also decreases with distance. Since intensity is proportional to 1 2 the square of the amplitude, the amplitude A must decrease as 𝑟 so that 𝐼 ∝ 𝐴 will be 1 proportional to 2. Speaking algebraically, the formula can explain what is meant by 𝑟 this: 1 𝐴∝ 𝑟 , and thus 𝐴2 𝑟1 𝐴1 = 𝑟2 Reflection, Refraction, Diffraction, Interference, Superposition When a wave encounters an object, the wave is either transmitted, reflected, absorbed, refracted, polarized, diffracted, or scattered depending on the wavelength and material composition of the object. 1. Reflection is defined as the change of direction of propagation of a wave when it meets a boundary or barrier. In layman’s terms, the wave bounces off the object. The reflected light then goes in a different direction. All types of waves follow the law of reflection. The angle between the incoming light and a line perpendicular to the surface we call the angle of incidence. The angle between the reflected light and a line perpendicular to the surface we call the angle of reflection. The perpendicular line we call the normal.  FIGURE 2.4: Reflection of Light Diagram Always remember, that the Angle of Incidence and Angle of Light are ALWAYS EQUAL θ𝑖 = θ𝑟