Prove that using x(t) = Asin (ωt + ϕ) will produce the same results for the period for the oscillations of a mass and a spring. Why do you think the cosine function was chosen?
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Prove that using x(t) = Asin (ωt + ϕ) will produce the same results for the period for the oscillations of a mass and a spring. Why do you think the cosine function was chosen?
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- Write the equations that describe the simple harmonic motion of a particle moving uniformly around a circle of radius8units, with linear speed 3units per second.To be able to solve rectilinear problems with variable functions. A particle moves with harmonic motion over a path 10in long in 10sec. Find A, V and t when it has moved 5in from one end.An object attached to a spring vibrates with simple harmonic motion as described by the figure below. A coordinate plane is shown with t (s) on the horizontal axis and x (cm) on the vertical axis. A curve is shown to make one and a half complete oscillations along t. The curve begins at the origin moving with a steep slope. The curve is moving with increasing x and decreasing slope until it is horizontal and at its maximum at (1, 2). From (1, 2) the slope of the curve becomes negative and steadily decreases until it crosses the t-axis at (2, 0) with a steep negative slope. From (2, 0) the curve continues below the t-axis with increasing slope until it is horizontal and at its minimum at (3, −2). From (3, −2) the slope of the curve steadily increases until the curve crosses the t-axisat (4, 0) with a steep slope. From (4, 0) one oscillation is complete and the curve repeats the same pattern, decreasing slope until the maximum at (5, 2) and continuing decreasing slope until crossing the…
- An object attached to a spring vibrates with simple harmonic motion as described by the figure below. A coordinate plane is shown with t (s) on the horizontal axis and x (cm) on the vertical axis. A curve is shown to make one and a half complete oscillations along t. The curve begins at the origin moving with a steep slope. The curve is moving with increasing x and decreasing slope until it is horizontal and at its maximum at (1, 2). From (1, 2) the slope of the curve becomes negative and steadily decreases until it crosses the t-axis at (2, 0) with a steep negative slope. From (2, 0) the curve continues below the t-axis with increasing slope until it is horizontal and at its minimum at (3, −2). From (3, −2) the slope of the curve steadily increases until the curve crosses the t-axis at (4, 0) with a steep slope. From (4, 0) one oscillation is complete and the curve repeats the same pattern, decreasing slope until the maximum at (5, 2) and continuing decreasing slope until crossing…A body of mass m is suspended by a rod of length L that pivots without friction (as shown). The mass is slowly lifted along a circular arc to a height h. a. Assuming the only force acting on the mass is the gravitational force, show that the component of this force acting along the arc of motion is F = mg sin u. b. Noting that an element of length along the path of the pendulum is ds = L du, evaluate an integral in u to show that the work done in lifting the mass to a height h is mgh.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 is
- Function y 2(sin(4x – 7) – 4) , determine its amplitude, phase shift and period.(a) Show that the transformationQ = p + iaq, P = (p − iaq) / (2ia)is canonical and find a generating function. (b) Use the transformation to solve the linear harmonic oscillator problem.If a mass m is placed at the end of a spring, and if the mass is pulled downward and released, the mass-spring system will begin to oscillate. The displacement y of the mass from its resting position is given by a function of the form y = c,cos wt + c2 sin wt (1) where w is a constant that depends on spring and mass. Show that set of all functions in (1) is a vector space.
- (a) Show that the transformation Q = p + iaq, P = (p − iaq) / (2ia) is canonical and find a generating function. (b) Use the transformation to solve the linear harmonic oscillator problemA rope passes through a small hole made on a smooth horizontal table. Two particles of equal mass are tied one at each end of the string. One of the particles hangs vertically and the other remains on the table at a distance a from the hole. The particle on the table is projected with a velocity Vga perpendicular to the chord. Show that the hanging particle remains at rest. If the hanging particle is slightly disturbed in the vertical direction. Show that the period of the small oscillations is 2n/2a/3g.Consider the length of the simple pendulum is 1 and the mass of the pendulum bob is m. Obtain the equation for the Lagrangian for the simple pendulum when its kinetic energy is 1 m²o² and the potential energy is mgl(1-cos ø). 1 -ml²ở –mgl(1- cos ø) 1 (B) - mlo – mgl(1-cos ø) 1 (C) -m²o² + mgl (1– cos ø) 2 1 (D) ml o – 2mgl (1- cos ø)