Gravity and Periodic Motion PDF

SH1685 Gravity and Periodic Motion accelerations due to their gravities. Gravity is also present in tides, or tidal movement, celestial mechanics, and potential energy. The t...

SH1685 Gravity and Periodic Motion accelerations due to their gravities. Gravity is also present in tides, or tidal movement, celestial mechanics, and potential energy. The tide is I. Gravity high because our water rises in height, and the opposite applies in low We all have heard that story. There was a young man with a brilliant tide. mind, resting underneath a tree. Suddenly, its fruit fell and landed hard on the man’s head. He was surprised -- and pained, but his mind was Earth has interactions with the Sun and the Moon. According to piqued as to how this fruit always falls to the ground. From that Newton’s Third Law, one action is met with another action of equal moment onwards, that young man has changed the world forever. magnitude, locking them as one balanced system. Earth experiences that law with the Sun and the Moon. The Sun exerts its gravity on our It was Isaac Newton, and his story was somewhat of a flamboyant planet, pulling it. But, our planet pulls the Sun, as well. We simply do version of what happened, which was almost lost in history. Still, his not experience that pull from the Sun because we are on Earth. Plus, ideas, along with a few other natural philosophers (scientists in the the Sun is very far, allowing us to experience a fraction of the Sun’s modern age), have helped shape our understanding of Physics. actual gravity. The same goes for the Moon and Earth. As these go on with their interactions, the water on the Earth’s surface goes up Masking as Other Stuff because the water flows more freely, as seen in the image on the We all know that gravity (Lat. gravis, “heavy”) is a force that pulls opposite side. Therein exists a tidal bulge, which is the result of Earth’s everything towards its center. We are already experiencing its rotation, plus interlocking forces that affect the planet. influence, from the glass of milk spilling to the floor to our aching feet due to standing for too long or diving from a cliff into the deep blue There are two tides, illustrated by the image. Spring tides are the sea. We also know that the phrase, “What comes up, must go down,” highest of the high tides, as well as the lowest of the low tides, due to somehow is influenced by it. We also know that gravity is a force, just the positions of the three objects concerned. If the Sun, the Moon, and like a push or pull. the Earth are aligned perfectly, the water rises due to the strong interaction between the three objects. Even if the Moon blocks the But, gravity is more of a pull than a push. Actually, eliminate the push Sun, there still exists a spring tide on the side not being pulled by the out of your mind whenever we talk about gravity because it will never Moon (i.e., the far side of the Earth) because of the tidal forces our push things away. That is the true nature of gravity. planet is experiencing. To define tidal force simply, it is a force that results from the gravitational pulls of both Earth and the Moon, which An object has gravity because it has mass. The heavier an object is, causes the water on the far side to “move” away from the planet the higher its gravity, even if two (2) objects have the same size. That because the Moon is pulling the Earth towards it! is because one object has more stuff (i.e., mass) than the other. The other has less stuff; therefore, it has a lower influence than the other. The neap tides are the tides with the small variances from both low But, gravity is experienced in a variety of ways. For one, it is a force, and high (i.e., almost no difference) tides. It only occurs when the Sun which we call gravitational force, or the force of gravity (𝐹𝑔 ). It is is perpendicular to the Earth-Moon alignment. Other celestial bodies responsible for giving us our Newtonian weight (𝐹𝑔 = Weight), which also experience this, with black holes as examples. Here, the tidal is calculated through the constant, 𝑔. forces are so strong; black holes can tear matter from a nearby star. The constant for the acceleration due to gravity computed to be at Another thing is potential energy, or more specifically, gravitational 9.81 m/s 2 , which is shortened to 9.8 m/s 2 and is only applicable to potential energy (𝑈𝑔 ). We learned that, Earth. All other objects in space have varying forces of gravity and 𝑼𝒈 = 𝒎𝒈𝒚 (𝒎𝒈𝒉 𝐢𝐧 𝐨𝐭𝐡𝐞𝐫 𝐫𝐞𝐟𝐞𝐫𝐞𝐧𝐜𝐞𝐬), 09 Handout 1 *Property of STI  [email protected] Page 1 of 10  SH1685 energy is the energy stored with respect to the Earth’s gravitational field. But, if there were to be some changes involved, then, ∆𝑼𝒈 = 𝒎𝒈∆𝒚. So, if the final height is greater than the initial height, the potential energy is positive. Likewise, the value becomes negative once the initial height becomes higher than the final height. Newton’s Law of Universal Gravitation Gravity is a pulling force that keeps objects as small as an apple falling to the ground or as large as two galaxies to interact with one another. But, how do we know that gravity is also responsible for the movements of celestial bodies? This topic will tackle one of nature’s fundamental laws -- the Law of Universal Gravitation. The center of mass is the point where all mass is concentrated. That is also where gravity also influences that object, which is called the center of gravity. The center of mass is always used in larger bodies because it is more convenient to do so. By applying the center of masses between two (2) objects, Newton’s Law of Universal where 𝑚 = mass of an object Gravitation possesses the formula, 𝑔 = acceleration due to gravity 𝑦 = height of the object from the ground (𝒎𝟏 𝒎𝟐 ) 𝑭𝒈 = 𝑮 , 𝒓𝟐 From that equation alone, we also learned that energy has the same inclination as mechanical work, where 𝐹𝑔 = force of gravity 𝑚 = mass of an object 𝑊 = 𝐹𝑥, 𝑟 = distance between the objects (or radius) 𝐺 = gravitation constant where 𝑥 = distance m3 𝐹 = force (net force) where 𝐺 = 6.673 × 10−11. kg ∙ s 2 = (𝑚𝑎) Newton theorized that planetary motion was in conic sections – Applying analogy, 𝑚𝑔 from 𝑈𝑔 is the same as 𝑚𝑎 from the net force, circular, ellipsoid, parabolic, and hyperbolic shapes. That is why and both 𝑥 and 𝑦 are both displacements. So, gravitational potential distance in this formula is treated as a radius. 09 Handout 1 *Property of STI  [email protected] Page 2 of 10  SH1685 According to this Law: in which if the system is truly conservative, then ∆𝑇 + ∆𝑈 = 0, which 1. The greater the mass of one (1) object, the greater the force of in transposition, we get that, gravity it exerts over the other object. 2. The farther the two (2) objects are from one another, the weaker ∆𝑇 = −∆𝑈. the force of gravity. If we substitute in the proper values correlating to gravity, If we insert the conservative work equation, which is equivalent to weight multiplied by a factor of one (1), which denotes that the object 1 𝑚1 𝑚2 is at ground level, 𝑚𝑣 2 = −𝐺. 2 𝑟 𝑊 = 𝑚𝑔𝑦 𝑚1 𝑚2 We get that value for 𝑈 because, 𝐹𝑔 = 𝐺 , 𝑟2 𝑚1 𝑚2 since 𝑊 = 𝑚𝑔, with 𝑦 = 1, and 𝐹𝑔 = 𝑚𝑔, 𝐹𝑔 = 𝐺. 𝑟2 𝑚1 𝑚2 If 𝐹𝑔 = 𝑚𝑔, then, 𝑚𝑔 = 𝐺. 𝑟2 𝑈 = −𝐹𝑔 𝑦, We can also use the equation to determine the strength of an object’s since gravitational field (𝑔𝐹 ), 𝑚1 𝑚2 1 𝑚1 𝑚2 1 𝑚𝑔 = −𝐺 , [𝑚𝑔 = 𝐺 ] 𝑟2 𝑚 𝑟2 𝑚 𝑚1 𝑚2 𝑚 𝑈 = − (𝐺 ) 𝑦. 𝑔𝐹 = 𝐺 2. 𝑟2 𝑟 Since 𝑦 = 𝑟, which are variables for distance, and applying elimination, Newton’s Law of Universal Gravitation stated that the farther the two (2) objects are from one another, the weaker the gravity. It is such 𝑚1 𝑚2 gravity that astronauts experience once they are in outer space. They 𝑈 = − (𝐺 )𝑟 {𝑟}{𝑟} are within Earth’s influence, yet at the same time, somehow 𝑚1 𝑚2 = −𝐺. experience some degree of freedom. This is called microgravity, in 𝑟 which the gravitational influence is so weak that the one experiencing it has that sensation of weightlessness. Also, the gravitational field If we try to solve for 𝑣, we get that, value (𝑔𝐹 ) has almost the same value as the acceleration due to gravity. 1 𝑚1 𝑚2 𝑚𝑣2 = − (−𝐺 ) 2 𝑟 Escape Velocity 1 𝑚 1 𝑚2 Both potential and kinetic energy is required to perform conservative 𝑚𝑣2 = 𝐺 2 𝑟 work within a system, 𝑚1 𝑚2 𝑊 = ∆𝑇 − ∆𝑈 𝑚𝑣 2 = 2𝐺 𝑟 09 Handout 1 *Property of STI  [email protected] Page 3 of 10  SH1685 𝑚 𝑣2 = 2𝐺 Kepler’s First Law 𝑟 𝟐𝑮𝒎 𝒗=√. 𝒓 This equation is the equation for escape velocity (𝑣𝑒 ), which is the speed required by any object to “escape” or break free, from a body’s gravitational influence, specifically for celestial bodies. It applies to any object, even small particles, such as gas molecules. As such, rewriting this equation, 𝟐𝑮𝒎 𝒗𝒆 = √. 𝒓 Kepler’s Laws of Planetary Motion As theorized by Newton, planets and other celestial bodies have their Source: http://hyperphysics.phy-astr.gsu.edu paths shaped like conic sections. This theory was eventually proven by Johannes Kepler, a mathematician who helped the astronomer 2. The Law of Equal Areas. This states that all calculated areas, Tycho Brahe in his astronomical research. Kepler’s calculations. His which result from a planet’s movement around the Sun, are all computations are named by astronomers as “Kepler’s Laws of equal in values. Planetary Motion”. Kepler’s Second Law The three (3) Laws of Planetary Motion are: 1. The Law of Elliptical Orbits. Also known as the Law of Ellipses, it states that all orbital paths of planets take on an elliptical shape, with the Sun as one of its focus. If we are to recall our geometry, an ellipse is a conic section wherein it has foci located in its semi- major axis. You can check the diagram on the next page. Source: https://eloisechen.files.wordpress.com/ This is best explained by the astronomical terms perihelion and aphelion. Perihelion is the period of a planet wherein it is close to the Sun, thus creates a very wide triangle, or “pizza slice”, colloquially. The aphelion, on the other hand, is the farthest point 09 Handout 1 *Property of STI  [email protected] Page 4 of 10  SH1685 of a planet with respect to the Sun, thereby creating a very narrow The period (T) is the amount of time it takes for the motion to repeat “pizza slice”. They may look different, but they have the same itself, much like it is the measured time for an object to completely area. move around its designated center. Frequency (f) is defined to be the number of events per unit of time. For periodic motion, frequency is 3. The Law of Harmonies. This law states that, by observing the the number of oscillations per unit time. The relationship between motions of two planets moving around the Sun, therein lies a frequency and period is, calculated ratio where the ratio of the squares of the planets’ 𝟏 𝒇=. periods (𝑇) is equal to the ratio of the cubes of the planets’ average 𝑻 distances (𝑟) from the Sun. In Mathematical terms, The unit of period (T) is second (s) or any unit of time. The unit of (𝑻𝟏 )𝟐 (𝒓𝟏 )𝟐 frequency is s–1, "/s", or any reciprocal unit of time. s–1 is commonly =. known as hertz (Hz), where (𝑻𝟐 )𝟐 (𝒓𝟐 )𝟐 This Law only provides the descriptive equation for the objects’ 𝟏 𝑯𝒛 = 𝟏 𝐬 –𝟏. movements, not the actual definition as to why there is harmony between them. It is only valid to two (2) smaller objects revolving Objects attached to a spring or any elastic object with negligible mass around a larger body. is governed by Hooke’s Law. Note that this is the same Hooke’s Law from where we have discerned the Elastic Potential Energy, While Kepler’s Laws are mostly used by astronomers, these are also valid for small-scale orbiting bodies, such as natural and artificial 𝑭𝒓 = −𝒌𝒙, satellites, among others. We call the study of the movement of the heavenly bodies celestial mechanics. where Fr is the restoring force, k is the spring constant and x is the displacement from the spring’s relaxed length. The restoring force is II. Periodic Motion always opposite the displacement (hence, the negative sign) and also Suppose we have a round marble moving within a circular path. If we directly proportional to the magnitude of the displacement. This keeps are to look above, we see that it is indeed moving in a circle. But, if we the object in moving back and forth. The diagram below shows the are to look at its side, it appears to be moving from left to right. So, time (t) of a block oscillating at the end of spring. what is it? That movement is called periodic motion, wherein it repeats its movements. Examples are swings at the playground, a coil spring, and even a wave! It is sometimes called harmonic motion because it has “harmony”, where scientists use the word in place of pattern, repetition, recurrence, etc. There is a key difference, however. Periodic motion can also refer to the movements in a conic section that repeats itself over a certain timeframe, whereas harmonic motion only applies to partly- to non-circular movements, like swinging. 09 Handout 1 *Property of STI  [email protected] Page 5 of 10  SH1685 Periodic graphs, no matter how tall or short, how wide or narrow, or whether the function value at t = 0 is zero, can be represented by a sine curve. One can express the position as a function of time as, 𝒙(𝒕) = 𝒙𝟎 𝐬𝐢𝐧(𝝎𝒕 + 𝜹). In the given equation above, t is the time (independent variable), and x is the displacement (dependent variable). The parameters xo, ω, and δ are constants that respectively show the extreme values of the function 𝑥(𝑡), the width of the graph, and the shift of the graph from the actual sine function. Take note that sin 𝜔𝑡 + 𝛿 has no unit and can range only from –1 to 1. Hence, the unit of xo is the meter (or any unit of distance). The expression 𝜔𝑡 + 𝛿 is an angle. Therefore, the unit of δ must be δ rad, and ω must be ω rad/s. The variable xo is called amplitude (represented as 𝐴 in other references); the magnitude of the maximum and minimum displacement. The parameter ω is called angular frequency. Angular frequency is a parameter that describes the vibration and does not involve an actual motion or rotation. The period of the harmonic motion depends on angular frequency. The period is related to angular frequency by the formula, 𝟐𝝅 𝑻=. 𝝎 Meaning, if ω is smaller, T is larger, and the graph appears wider. If ω is larger, T is smaller, and the graph appears narrower. The variable δ is called the phase angle. It represents the shift of the graph from If we graph the position of the object undergoing simple harmonic the standard sine function where the function value at 0, δ, 2δ, 3δ, …, motion (such as a mass attached to a spring), with time as the nδ (n = any integer) are 0. If δ is positive, the graph is shifted to the independent variable, we’ll get a graph similar to the image below. left. If δ is negative, the graph is shifted to the right. The displacement is not the only quantity that fluctuates between two extremes of the same magnitude but opposite directions. Velocity also fluctuates between maximum velocity in one direction and maximum velocity in the opposite direction. However, displacement and velocity never reach their maximums at the same time. Rather, if the velocity is at maximum, the displacement is zero. If the displacement is at maximum, the velocity is zero. 09 Handout 1 *Property of STI  [email protected] Page 6 of 10  SH1685 Recalling the image earlier, it shows that the distance fluctuates at In it, we also have discussed that the energy equation for this scenario certain intervals. The velocity of an oscillating object also fluctuates, is, but it fluctuates differently. Below is the table that shows the intervals 𝐸 = 𝑇 + 𝑈𝑒 and its respective fluctuations, 1 1 = ( 𝑚𝑣 2 ) + ( 𝑘𝑥 2 ). 2 2 Time Velocity 𝑡 = 0 𝑣=0 Since we know that, in the scenario above, the amplitude is also equal 1 to the displacement, 𝑡 = 𝑇 𝑣>0 1 1 8 𝐸 = ( 𝑚𝑣 2 ) + ( 𝑘𝑥02 ). 2 2 1 𝑡 = 𝑇 𝑣 = 𝑣𝑚𝑎𝑥 4 In the case of the turning points, the velocity is equal to zero (0), 3 1 1 𝑡 = 𝑇 𝑣 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠 0 8 𝐸 = ( 𝑚2 ) + ( 𝑘𝑥 2 ) 2 2 1 1 2 𝑡 = 𝑇 𝑣=0 = 𝑘𝑥0. 2 2 5 If we are to include the maximum velocity, which also happens to be 𝑡 = 𝑇 𝑣 > 0, opposite 8 the object’s equilibrium point (due to the change of potential energy 𝑈 3 to kinetic energy 𝑇), 𝑡 = 𝑇 𝑣 = 𝑣𝑚𝑎𝑥 , opposite 1 4 𝐸 = 𝑚𝑣𝑚𝑎𝑥 2. 2 7 𝑣 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠 0, 𝑡 = 𝑇 8 opposite From here, we can compute for the object’s maximum velocity, 𝑡 = 𝑇 𝑣=0 1 2 1 𝑚𝑣𝑚𝑎𝑥 = 𝑘𝑥02 𝑚 At each oscillation where 𝑣 = 0, it shows that the spring has stored 2 2 2 enough elastic potential energy (𝑈𝑒 ) to propel itself. Notice that 𝑣 = 0 𝑣𝑚𝑎𝑥 = 𝑘𝑥02 has occurred three (3) times. This goes to show that the object first 2 𝑘𝑥02 𝑣𝑚𝑎𝑥 = began at its first turning point, storing 𝑈𝑒. As the spring is released, it 𝑚 moves until it reaches its second turning point; from there, the spring 𝑘𝑥02 restores itself from the built-up 𝑈𝑒 , going back to its first turning point, 𝑣𝑚𝑎𝑥 = √ repeating the cycle. 𝑚 𝒌 Knowing this, we can safely say that the object reaching its maximum 𝒗𝒎𝒂𝒙 = ± 𝒙𝟎 √. velocity (𝑣𝑚𝑎𝑥 ) is releasing its kinetic energy, serving as the harmonic 𝒎 movement’s equilibrium. From there, we can assume that the distance between the two turning points is the amplitude. 09 Handout 1 *Property of STI  [email protected] Page 7 of 10  SH1685 The sign is either positive or negative, which is determined by the angular position of the pendulum at a certain time t can be computed direction the object is heading to. If we are to treat velocity as a using, function of time, however, we can determine the instantaneous 𝜽(𝒕) = 𝜽𝟎 𝐬𝐢𝐧(𝝎𝒕 + 𝜹), velocity of the object instead. The instantaneous velocity of the object in simple harmonic motion can be computed using the formula, where 𝜃0 , 𝜔 and 𝛿 are constants. Similar to the previous equation 𝒗(𝒕) = 𝝎𝒙𝟎 𝐜𝐨𝐬(𝝎𝒕 + 𝜹). for the displacement-time function, 𝜃o is the amplitude of The acceleration of an object in simple harmonic motion also the harmonic motion, 𝜔 is the fluctuates between two (2) extremes. Displacement and acceleration angular frequency, and 𝛿 is the are always zero at the same time. However, whenever displacement phase angle. Note that 𝜔 and 𝛿, is at the positive extreme, acceleration is always at the negative like in the displacement-time extreme, and vice-versa. It is because force and acceleration are function equation, are merely always directed in the same direction according to Newton’s second parameters of harmonic motion law, while according to Hooke’s law, displacement and force are and not actual angles like 𝜃 and always towards opposite directions. The acceleration of an object in 𝜃o. Hence, in angular simple simple harmonic motion can be computed using the formula, harmonic motion, 𝜔 is not equal to, 𝒂(𝒕) = −𝝎𝟐 𝒙𝟎 𝐬𝐢𝐧(𝝎𝒕 + 𝜹). Source: http://img.sparknotes.com/content/testprep/booki mgs/sat2/physics/0011/pendulum.gif 𝜽 𝜽𝟎 Since the amplitude from the displacement-time function 𝑥(𝑡) = 𝐨𝐫. 𝒕 𝒕 𝑥0 sin(𝜔𝑡 + 𝛿) can be incorporated with the acceleration equation above; we can then simplify it into, The equation for the period (𝑇), 2𝜋 𝒂(𝒕) = −𝝎𝟐 𝒙(𝒕). 𝑇= , 𝜔 Pendulum still holds for both pendulums and springs. Pendulums undergo While some objects undergoing simple harmonic motion move back angular harmonic motion, whereas springs undergo spring harmonic and forth as governed by Hooke’s law, some objects rotate back and motion. There are three (3) types of pendulums: simple, physical, and forth between two angular positions. These objects are generally torsion pendulums. called pendulums, whereas the objects that undergo Hooke’s Law are called springs. Take, for example, a simple pendulum that moves back A simple pendulum is a system consists of a particle of mass m (called and forth at a certain angle, 𝜃. In its motion lie two equal angles, which the bob of the pendulum) suspended from one end of a massless can be shown by bisecting the angle itself. What lies between are two string of fixed length L that is attached at the other end. The bob is angles, 𝜃0 and −𝜃0. These two angles can be demonstrated below. free to swing back and forth in the plane of the page, to the left and right of a vertical line through the pendulum’s pivot point. To solve for As you can see in the image on the next page, 𝜃0 and −𝜃0 are equal, the angular frequency and period, the following formulas are, and are the maximum angles a pendulum can attain. From there, the 09 Handout 1 *Property of STI  [email protected] Page 8 of 10  SH1685 𝒈 With enough twisting, the spring pushes the weight in the opposite 𝝎=√ direction, creating an angular oscillation. To solve for the angular 𝑳 frequency and period of a torsion pendulum, the following formulas 𝑳 are, 𝑻 = 𝟐𝝅√. 𝒈 𝒌 𝝎=√ where, 𝒊 𝐿 = length of the massless string or rod 𝒊 𝑔 = acceleration due to gravity 𝑻 = 𝟐𝝅√ , 𝒌 Examples of a simple pendulum are the playground swing and the metronome. where, 𝑖 = object ′ s moment of inertia If a pendulum has two (2) pivots, it is considered to be a physical 𝑘 = spring constant of the rod. pendulum. In a physical pendulum, one (1) of the two (2) pivots is the main pivot, while the second pivot is located at the pendulum’s center If the period (𝑇) is not identified, we can make a simple analogy, going of mass. This causes the bob to swing at a smaller angle and cannot back to rotational motion. The period in rotational motion has the pass through the center of mass itself, unlike the simple pendulum. To equation, solve for the angular frequency and period of a physical pendulum, the 𝐶 𝑇= following formulas are, 𝑣𝑚𝑎𝑥 𝒎𝒈𝒓 The amplitude (𝑥0 ) is equal to the circle’s radius (𝑟), which is also 𝝎=√ 𝒊 equivalent to displacement (𝑥). With this, the circumference is equal 𝒊 to, 𝑻 = 𝟐𝝅√ , 𝐶 = 2𝜋𝑥0. 𝒎𝒈𝒓 Combining both equations, we get that, where, 𝑚 = mass of the bob 2𝜋𝑥0 𝑇=. 𝑖 = object ′ s moment of inertia 𝑣𝑚𝑎𝑥 𝑔 = acceleration due to gravity 𝑟 = length between the main pivot and the object′s center of mass Since we also know the equation for maximum velocity (𝑣𝑚𝑎𝑥 ), Examples of a physical pendulum are the bell and the human arm. 𝑘 𝑣𝑚𝑎𝑥 = 𝑥0 √. The third pendulum has a unique take. Its rod is fixed and cannot 𝑚 swing. Instead, the pendulum twists back and forth, creating an angular harmonic motion. This is the torsion pendulum. It is commonly Combining both equations, we get, used in torsion clocks, where the weight rotates freely around a spring. 09 Handout 1 *Property of STI  [email protected] Page 9 of 10  SH1685 2𝜋𝑥0 References: 𝑇= Bauer, W. & Westfall, G. D. (2016). General physics 1 (2nd ed.). 𝑣𝑚𝑎𝑥 2𝜋𝑥0 Columbus, OH: McGraw-Hill Education. = Bauer, W., & Westfall, G. D. (2016). General physics 1 (2nd ed.). Quezon 𝑘 City: Abiva Publishing House, Inc. 𝑥0 √ 𝑚 Bautista, D.C. (2013). Science impact: Integrated science (3rd ed.). 1 Antipolo City: Academe Publishing House, Inc. = 2𝜋 Belleza, R.V., Gadong, E.S.A., …, Sharma, M. Ph.D. (2016). General √𝑘 physics 1. Quezon City, Vibal Publishing House, Inc. 𝑚 𝒎 Catchilar, Gerry C. & Malenab, Ryan G. (2003). Fundamentals of physics. 𝑻 = 𝟐𝝅√. Mandaluyong City: National Book Store. 𝒌 Cordero-Navaza, Delia & Valdez, Bienvenido J. (2006). Physics IV (2nd where, ed.). Quezon City, Phoenix Publishing house, Inc. Freedman, R. A., Ford, A. L., & Young, H. D. (2011). Sears and 𝑚 = object ′ s mass Zemansky's University Physics (with Modern Physics) (13th ed.). 𝑘 = spring constant. Addison-Wesley. Giambattista, A., Richardson, B. M., Richardson, R. C, (2007). College So, if we are to determine the frequency we get, physics (2nd Ed.). New York: The McGraw-Hill Companies, Inc. Halliday, D., Resnick, R. & Walker, J. (2007). Fundamentals of physics 1 (5th ed.), New York: John Wiley and Sons, Inc. 𝑓= 𝑇 Hewitt, Paul G. (2007). Conceptual physics (3rd ed.). California: Addison- 1 Wesley Publishing Company = 𝑚 Nave, C. R. (2016). Circular motion. Retrieved from The Georgia State 2𝜋√ University’s HyperPhysics: http://hyperphysics.phy- 𝑘 astr.gsu.edu/hbase/circ.html#circ 𝟏 𝒌 The Physics Classroom. (2016). Circular motion and satellite motion. 𝒇= √. 𝟐𝝅 𝒎 Retrieved from The Physics Classroom website: http://www.physicsclassroom.com/class/circles Plait, P. (2015). The Gravity of the situation: Crash course astronomy #7. And, to determine the angular velocity quickly, it is simply, Retrieved from YouTube: https://www.youtube.com/watch?v=TRAbZxQHlVw 𝒌 Santiago, K. S. & Silverio, A. A. (2016). Exploring Life Through Science: 𝝎=√. Senior High School Physical Science. Quezon City: Phoenix 𝒎 Publishing House, Inc. Somara, S. (2016). Newtonian gravity: Crash course physics #8. Retrieved from YouTube: https://www.youtube.com/watch?v=7gf6YpdvtE0 Wilson, Jerry D. & Buffa, Anthony J. (2003). Physics (4th ed.). Prentice- Hall Perfect symmetry, Bantam Books 09 Handout 1 *Property of STI  [email protected] Page 10 of 10