Simple Harmonic Motion Lab-2

1. Consider a harmonic oscillator whose elastic potential energy is equal to its kinetic energy. Given that its angular frequency is w and its amplitude A, what is the magnitude of its velocity? 2. An object with mass m has moment inertia I about its center of mass. If we take the moment of inertia at a distance 3d from the center of mass, how does I change?

2. A simple pendulum consists of a point mass m suspended by a massless string with length. The point mass will oscillate back and forth along a circular are with radius . This is an example of a system that approximately undergoes simple harmonic motion under certain conditions. m a) Determine the angular acceleration of the point mass, which should lead you to a differential equation for the angle 0. b) Now suppose the angle is very small. If the angle is small, sin ≈ 0 (if you look at a graph of sin 0, for small 0, you should see that sin is approximately linear). Using this approximation, show that your answer to part a) leads to simple harmonic motion. Find the period, frequency, and angular frequency of the oscillations. c) For this part, suppose l = 0.5 m, m = 2 kg, and at t=0 the mass is at rest at an angle = -0.1 rad (note: a negative angle mean's the mass is to the left of the vertical). What is as a function of time? d) Assuming the oscillations are small, what physical…

The disk rolls without slip a. What is the critical damping coefficient, c for the system? b. Plot the response of the system when the center of the disk is displaced 5 mm from equilibrium and released from rest, if: i. c= Cc/2 r= 40 cm 4 kN/m thin disk of mass m = 1 kg, no slip

Simple Harmonic Motion The displacement of a body moving, in the y-direction, with simple harmonic motion is given by: Y=A cos (ot) Where A is the amplitude (in metres), o second) and t is the elapsed time (in t). is the angular velocity (in rads per (a) Derive a time-independent expression for the velocity of this body in terms of displacement. (b) Derive a time-independent expression for the acceleration of this body in terms of velocity. You should use the symbol A for the amplitude and a for the acceleration.

Chapter 15, Problem 072 A uniform circular disk whose radius R is 14.7 cm is suspended as a physical pendulum from a point on its rim. (a) What is its period? (b) At what radial distance r < R is there a pivot point that gives the same period? (a) Number Units (b) Number Units Click if you would like to Show Work for this question: Open Show Work

Two pendula are shown in the figure. Each consists of a solid ball with uniform density and has a mass M. They are each suspended from the ceiling with massless rod as shown in the figure. The ball on the left pendulum is very small. The ball of the right pendulum has radius 1/2 L. L = 2.9 m a)Find the period T of the left pendulum for small displacements in s.  b) Find the period T of the right pendulum for small displacements in s.

A uniform disk of radius R = 0.3 meters and mass M = 0.8 kg can oscillate in the vertical plane, around an axis that passes through the pin, indicated in the figure, which is located at a distance “d” from the center of the disk. What is the value of “d” so that the period of oscillation is minimum?

A spring in vertical position is bolted into a fixed beam. When the spring is loaded with a 120 g object causing the spring to be displaced from its equilibrium position to 10 cm. It was loaded with a 150 g object then compressed upwards. Neglecting air resistance and damping, the object-spring assembly was subjected to oscillating motion. Calculate the following for an amplitude of 35 cm.a. The angular frequency in rad/s.b. The Period and Frequency in seconds and hertz, respectively.c. The Maximum Velocity in m/sd. The Maximum Acceleration in m/s^2e. The restoring force of the spring in N

This procedure has been used to "weigh" astronauts in space: A 42.5 kg chair is attached to a spring and allowed to oscillate. When it is empty, the chair takes 1.30 s to make one complete vibration. But with an astronaut sitting in it, with her feet off the floor, the chair now takes 2.54 s for one cycle. Part A What is the mass of the astronaut? For related problemsolving tips and strategies, you may want to view a Video Tutor Solution of Angular frequency, frequency, and period in shm. Express your answer with the appropriate units. µA ? mastronaut = Value Units

A pendulum is swinging back and forth. The angle A that the pendulum's string makes with the vertical line can be modelled by the function 0(t) = 0.15 cos(6t) a. Determine the maximum angular speed of the pendulum? When does it happen in its first oscillation? b. What is the maximum acceleration of the pendulum? When does it happen in its first oscillation?

Part A Two identical thin rods, each of mass m and length L, are joined at right angles to form an L-shaped object. This object is balanced on top of a sharp edge (Figure 1). If the object is displaced slightly, it oscillates. Assume that the magnitude of the acceleration due to gravity is g Find w, the angular frequency of oscillation of the object. Your answer for the angular frequency may contain the given variables m and L as well as g. • View Available Hint(s) Templates SymBols undo redo fese keyboard shortcuts Help Submit Previous Anawere X Incorrect; Try Again; 4 attempts remaining

1. Refer and use figure 1 and 2. Calculate the magnitude of force needed to displace the block by 12.0 cm. What is the maximum kinetic energy this object can attain? What is its maximum potential energy? What is the amplitude? Where you would find the maximum and minimum potential and kinetic energies? If the angular frequency is 10.0 rad-sec¹, calculate the mass of the object. Force (N) 6.00 5.00 4.00 3.00 2.00 1.00 0.00 0.00 Figure 1: Hooke's Constant Plot 1.00 3.00 Displacement (cm) 2.00 4.00 5.00

Which of the following statements is FALSE for a particle undergoing simple harmonic motion? Explain why.A. The restoring force is directly proportional to the displacement from equilibrium. B. The acceleration and velocity are always in opposite directions. C. The motion of the particle is always periodic. D. The graph of the acceleration against the displacement is a straight line with a negative slope.

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a. What is the translational velocity of the bottom tip of the pendulum at the moment that gravitational potential energy is 50% of its maximum? b. What effect would doubling the mass and length of the physical pendulum have on the answer to part (a) of the problem? c. Draw graphs of angular acceleration, tangential translational acceleration, and centripetal acceleration as functions of the instantaneous angle that the pendulum makes with the vertical. In all three graphs show the behavior of the acceleration from release with theta =38.4 degree until the pendulum is vertical and theta =0 degree.

Chapter 15, Problem 057 The amplitude of a lightly damped oscillator decreases by 1.5% during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle? Number Units the tolerance is +/-5% Click if you would like to Show Work for this question: Open Show Work

Pendulum Motion I. The motion of a simple pendulum, consisting of a mass M at the end of a rod of length L., is described by the following first- order differential equation: do _ - g sin 0 d0 where w= angular velocity (rad/s) 0 = angle of displacement from equilibrium position g = 9.81 m/s² L = 1.0 m Calculate the angular velocity of the pendulum beginning with the initial conditions 0= 10°, a w= 0.3.

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Find the torque due to gravity on the pendulum about its pivot. Find the torque due to tension on the pendulum about its pivot. Why does the pendulum exhibit oscillatory motion? Explain. L m

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A 2.4m uniform rod with a mass of 0.8kg acts as a physical pendulum. The rod is attached to a 3kg block a. What is the total inertia of the rod and the block? b. What is the period of this pendulum?

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A Read aloud Draw PROBLEM 5: THREE PENDULA 1. m m Three pendula of equal mass m and equal string length l are su spended from the ceiling. The first pendulum is a solid sphere, the se con d pendulum is a hollow sphere, and the third pendulum is a small mass (which you can treat as a point mass). HOMEWORK PROBLEMS FOR WEEK # 11 A) Which of the three pendula, if any will experien ce the largest torque (assume angle of oscillation is the same for all three)? B) Which pendulum, if any, will have the greatest oscill ation frequen cy?

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Q3: State whether the following statement is True or False 1 If P₁ + P₂ = constant, then the net external force equals zero. 2 When a particle moves in a conservative field of a force; potential energy remains constant. 3 Terminal velocity; only total force on the body is zero. 4 For any harmonic oscillator; it's true that: A² == Amax 5 The particle moving in a central field; angular momentum is constant. 6 One dimensional motion of a particle is always conservative if; force is a function of position.

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Problem2: An automobile suspension system is critically damped, and its period of free oscillation with no damping is 1 s. If the system is initially displaced by an amount and released with zero initial velocity (at 1 = 0; x(0) =x, and v(0) = 0) a- Determine the value of the angular frequency of free oscillation with no damping. (w, = ??) b- Deduce the value of friction factor (g =??) c- Write the expression of position as a function of time of the system, if the system is critically damped. (x(t) = ??) d- Deduce the expression of the velocity of the system. (v(f) = ??)

5A = Material point of mass m moves under the influence of force F-kr = –krî With in other words, the mass m is at the tip of an isotropic harmonic oscillator with equilibrium position at the origin of the axes. a) Calculate the potential energy V(r) of m. b) To design qualitatively 1) the potential energy V(r) of the mass m, 2) its "centrifugal" dynamic energy (r) = 1² /2mr² where L is the measure of angular momentum of the mass m and r its distance from the origin of the axes, and 3) the active potential energy of U(r) = V (r)+ Vä(r). "