Could you please answer the 'pre-laboratory questions' and show your work for any question you answer? here is the prelab question you have to answer:

Exp. M9: Simple Harmonic Motion Introductory Material "Simple harmonic motion" occurs when an object is subjected to a restoring force that is linearly proportional to the displacement of the object. An example of this is a spring with a spring constant k. The force F acting on the mass is F = -kx (9.1) where x is the displacement of the mass from equilibrium (x = 0). The spring applies this force both when it is stretched (positive x) and when it is compressed (negative x). The resulting motion of the object is sinusoidal x (1) = Asin(cot + $) (9.2) where A is the amplitude, @ is the angular frequency, and o is a constant called the phase. The angular frequency @ is inversely proportional to the period of the motion T, @ = 2TIT (9.3) For a spring and mass, the angular frequency is W = Vk/m (9.4) where m is the total mass of the glider + 1/3 of the mass of the spring Sinusoidal motion, x(t) = Asin(wt + $) Here, A is the amplitude and T is the period of the oscillation. The angular frequency is W= 2TIT. Position Because the time t = 0 is arbitrary, we have included a constant o, called the phase. Enbowfed qidanoitalon ord anisenet ouiloba in wod mes. Motion of a Mass Connected to Two Springs We would like to study this motion using the air track, because it is effective in eliminating friction. We will use the glider as the mass, and we will attach springs to it. It turns out that it is more convenient to use two springs, as sketched below, rather than just one. With two springs connected to the same object, the restoring force is increased. When the spring on the left is compressed, the one on the right is stretched, and vice versa. As a result, the M9 -2 A arlat bria swol to VinesExp. M9: Simple Harmonic Motion forces applied by the two springs are always in the same direction. Both springs push the mass (a) A single spring, with spring Mass constant k1, attached at one end to a M mass M (b) Two springs, each attached to a a Mass mass M. The motion of the mass is like M that for a single spring, but with a Spring 1 Spring 2 spring constant k = k1 + k2 Spring 1 Spring 2 c) The air track's glider, in the center, is attached to two springs, Motion and force sensors which are fixed at the ends of the air track. word woll Figure 9-1. Setup, with two springs. The motion and force sensor is at the right of photo (c). back to its equilibrium position. Thus, the total force acting on the mass is F = -k x, where (01.0) k = k,+ k2 (9.5) Here, ki and k2 are the spring constants for the two individual springs. Initial Conditions and Simple Harmonic Motion Three variables appear in Eq. (9.2) for the motion of the oscillating glider: A, w, and p. The value of w is determined solely by the mass of the glider and the spring constants. On the other hand, the values of A and o are determined by the initial conditions. If you pull on the mass a certain distance, and then release it at a certain time, that will determine the initial conditions. Suppose you pull the glider to a displacement xo at time t = to, hold it still, and then release it. The glider will then oscillate about its equilibrium position, in the range-X, SXSX For comparison, the sine function in Eq. (9.2) oscillates between the values +A and -A. Thus, the amplitude in Eq. (9.2) will be A = xo. The phase o merely shifts the graph to the left or the right. M9 -3Exp. M9: Simple Harmonic Motion m buff reuq egnhege mold .polls Energy in the Simple Harmonic Oscillator If no frictional forces act upon the glider/spring system, then the total mechanical energy E of the glider will be conserved. Here, E is a sum of the glider's kinetic energy K and the potential energy U stored in the springs E = K + U (9.6) The glider's kinetic energy is given by K = 1mv2 (9.7) while the potential energy stored in the springs is given by U = 2k,x2+2 k2x2 (9.8) The total mechanical energy of the simple harmonic oscillator, Eq. (9.9), is thus E = =mv2 + 2 (K, + K2)x2 (9.9) The velocity is v = Ax/ At. Using Eq. (9.2) for x(t), it is possible to show that v(t) = Aw cos(wt + ) . Therefore, the kinetic energy is Imo? A2 cos? (wt + $) . We can now write the total energy in Eq. (9.12) as gniion sotol lefor and Hog roundiliups all of doed E = >ma2 42 cos2(ct + $) + 2(k1 + k2) 42 sin?(cot + $) (9.10) bivibru oval According to Eq. (9.4) and (9.5), the sum of the spring constants k = ki + k2 may be substituted for maz E = 1(k1 + k2) 42 cos?(ct +$) + sin?(cot+ $) (9.11 ) it sobily a aolfort Sill wh We can replace the term in square brackets with unity by using the trigonometric identity sin2 0 + cos 0 =1. Thus, the total energy in the simple harmonic oscillator is Jon qua E = 2 ( k, + K2)A2 (9.12) ulboo which depends only on the strength of the springs and the amplitude of the motion. can Get it is more convoyle with evo springs Showing this requires calculus. The velocity is the derivative of the position with respect to time. M9 -4Exp. M9: Simple Harmonic Motion Pre-Laboratory Questions 1. A 5.0-g mass is sandwiched between two springs with spring constants ki and k2. The mass is displaced 10 cm from its equilibrium position and makes sixteen complete oscillations in 1 s with no loss of mechanical energy. Calculate the period of the motion T, the angular frequency @, the sum of the spring constants ki + k2, and the mechanical energy E of the oscillator. 2. The experiment is repeated, this time displacing the mass only one centimeter from the equilibrium position. Calculate the angular frequency @ of the motion and the mechanical energy E of the oscillator. this to Sghut oil T .nousernoo ( Old sri cini Toroslab nottorn ord easter of the red gilder worle lliw messe odr lanenod gange - CM qua" bolling li gneguyagus orb nogo Ed (Home to bayonet s ce sgot atold Myery fal got ed Tenoigon guidging wol tooka sno dilwe we gorol ard Into M Equipment List Air Blower* Hanging Mass Set Air Track* Level Air Track Gliders* (Gold, Red, Blue) Logger Pro Software* Air Track Springs (2, 5-cm) Motion Detector* Computer* Styrofoam Sheet Force Sensor* * An asterisk indicates that an item is described in the Instrument Glossary