The gravitational force on a particle of mass m inside the earth at a distance r from the center (r < the radius of the earth R) is F = −mgr/R (Chapter 6, Section 8, Problem 21). Show that a particle placed in an evacuated tube through the center of the earth would execute simple harmonic motion. Find the period of this motion.
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The gravitational force on a particle of mass m inside the earth at a distance r from the center (r < the radius of the earth R) is F = −mgr/R (Chapter 6, Section 8, Problem 21). Show that a particle placed in an evacuated tube through the center of the earth would execute
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- A mass of 55 grams stretches a spring by 8 cm. (Note that this means the forces balance, and thus mg = kx where m = 55 grams is mass, g = 981 cm/s is acceleration due to gravity, k is the spring constant, and x = 8 cm is the displacement.) The mass is set in motion from this equilibrium position with an initial downward velocity of 23 cm/s, and there is no damping. Find the position u (in cm) of the mass at any time t (in s). (Assume that position is measured upward from the equilibrium position.) u(t) Find the frequency (in radians per second), period (in seconds), and amplitude (in cm) of the motion. Frequency is Period is Amplitude isConsider the simple pendulum: a ball hanging at the end of a string. Derive the expression for the period of this physical pendulum, taking into account the finite size ball (i.e. the ball is not a point mass). Assume that the string is massless. Start with the expression for the period T'of a physical pendulum with small amplitude oscillati T = 2π The moment of inertia of the ball about an axis through the center of the ball is Here, I, is the moment of inertia about an axis through the pivot (fixed point at the top of the string, m is the mass of the ball, g is the Earth's gravitational constant of acceleration, and h is the distance from the pivot at the top of the string to the center of mass of the ball. Note, this pre-lab asks you to do some algebra, and may be a bit tricky. I mgh Iball = / mr² TA certain oscillator satisfies the equation of motion: ä + 4x = 0. Initially the particle is at the point x = V3 when it is projected towards the origin with speed 2. 2.1. Show that the position, x, of the particle at any given time, t, is given by: x = V3 cos 2t – sin 2t. (Note: the general solution of the equation of motion is given by: x = A Cos 2t + B Sin 2t, where A and B are arbitrary constants)
- A block of wood is floating in water; it is depressed slightly and then released to oscillate up and down. Assume that the top and bottom of the block are parallel planes which remain horizontal during the oscillations and that the sides of the block are vertical. Show that the period of the motion (neglecting friction) is 2π ph/g, where h is the vertical height of the part of the block under water when it is floating at rest. Hint: Recall that the buoyant force is equal to the weight of displaced waterA particle of mass m, which is constrained to move along a curve in the vertical plane, performs simple harmonic oscillations with an amplitude-independent period T = 27 (1) where l is a constant length. Determine the curve s = s(0) on which the particle moves.Consider a simple harmonic oscillator with natural frequency, w = 1 and the displacement from equilibrium of the mass is denoted by x. At t=0, the mass is released from x=1m with speed 2m/s. What is the magnitude of x at time, t= s?
- The equation of motion for a damped harmonic oscillator is s(t) = Ae^(−kt) sin(ωt + δ),where A, k, ω, δ are constants. (This represents, for example, the position of springrelative to its rest position if it is restricted from freely oscillating as it normally would).(a) Find the velocity of the oscillator at any time t.(b) At what time(s) is the oscillator stopped?The graph shown in the figure (Figure 1) closely approximates the displacement at of a tuning fork as a function of time t as it is playing a single note. Figure x(mm) 0.4 0.2 -0.2 -0.4 AMA 2 (ms) 1 of 1 What is the amplitude of this fork's motion? A = Submit Part B T = What is the period of this fork's motion? Submit Part C | ΑΣΦ Request Answer | ΑΣΦ What is the frequency of this fork's motion? f = 426 Part D Request Answer ΠΑΣΦ Submit Previous Answers Request Answer w = 2π (224) What is the angular frequency of this fork's motion? ID ΑΣΦ ? * Incorrect; Try Again; 4 attempts remaining ? ? ? mm mns Hz rad/s PearsonA ball whose mass is 1.4 kg is suspended from a spring whose stiffness is 40N/m. The ball oscillates up and down with an amplitude of 14 cm. Suppose this apparatus were taken to the Moon, where the strength of the gravitational field is only 1/6 of that on Earth. What would be the period on the Moon? (Consider carefully how the period depends on properties of the system; look at the equation.)
- A pendulum has a period of 5.8 s. If you were to transfer this pendulum to the surface of Mars (where g is 3.71 meters per second squared), what would the period (in seconds) be?A cylindrical disc with a mass of 0.619 kg and radius of 0.575 m, is positioned such that it will oscillate as a physical pendulum as shown below. If the period of the small angle oscillations is to be 0.343 s, at what distance from the center of the disc should the axis of rotation be fixed? Assume that the position of the fixed axis is on the actual disc. The moment of inertia of a disc about its center is 1 = 0.5 M R²...Hint: Use the parallel axis theorem.A meter stick of total length l is pivoted a distance d from one end on a friction-less bearing. The stick is suspended so that it becomes a pendulum. This is called a "physical" pendulum because the mass is distributed over the body of the stick. Assume the total mass is m and the mass density of the stick is uniform. The acceleration of gravity is g. Find T and V as functions of generalized coordinate θ and velocity ˙θ. Do this by considering the stick to be divided into infinitesimal parts of length dl and integrating to find the total kinetic and potential energy. Set up the Lagrangian and find the equation of motion.