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A particle of mass m is at rest at the end of a spring (force constant = k) hanging from a fixed support. At t = 0, a constant downward force F is applied to the mass and acts for a time t0. Show that, after the force is removed, the displacement of the mass from its equilibrium position (x = x0, where x is down) is
where
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Chapter 3 Solutions
Classical Dynamics of Particles and Systems
- A hollow steel ball weighing 24 pounds is suspended from a spring. This stretches the spring 4 inches. The ball is started in motion from a point 3 inches above the equilibrium position. Let u(t) be the displacement of the mass from equilibrium. Suppose that after t seconds the ball is u feet below its rest position. Find u (in feet) in terms of t. (Note that the positive direction is down.) Take as the gravitational acceleration 32 feet per second per second. u=arrow_forwardIf 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.arrow_forwardA 0.175 kg mass on a table is attached to a horizontal spring with a spring constant of 155 N/m. Ignore all friction and air resistance in this problem (i.e. assume no non-conservative forces act on the system). Assuming you call time t=0 the time when the mass is moving to the left (in the negative direction) through equilibrium, sketch a graph of the horizontal displacement of the mass vs. time for two full periods. Make sure to label your axes correctly and to mark the numerical values of your amplitude and period on the graph. Determine whether the mass is located to the left or the right of equilibrium at t=0.240 s. Determine if the mass is moving to the left or to the right at t=0.240 s. At some time you make a measurement of the mass and find that it is moving at a speed of 0.815 m/s and is 0.03 m from equilibrium. Find the amplitude of the oscillation.arrow_forward
- An object of mass 3 grams is attached to a vertical spring with spring constant 27 grams/secʻ. Neglect any friction with the air. (a) Find the differential equation y" = f(y, y') satisfied by the function y, the displacement of the object from its equilibrium position, positive downwards. Write y for y(t) and yp for y' (t). y" : -9y Σ (b) Find r1, r2, roots of the characteristic polynomial of the equation above. r1,r2 = Зі, - 3і Σ (b) Find a set of real-valued fundamental solutions to the differential equation above. Y1(t) = cos(3t) Σ Y2(t) = sin(3t) Σ (c) At t = 0 the object is pulled down 1 cm and the released with an initial velocity downwards of 3/3 cm/sec. Find the amplitude A > 0 and the phase shift o E (-1, 7| of the subsequent movement. A = Σφ Σarrow_forwardA weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by y = 1/6 sin(8t) + 1/8 cos(8t) where y is the displacement (in feet) from equilibrium of the weight and t is the time (in seconds). (a) Use the identity a sin(Bθ) + b cos(Bθ) = a2 + b2 sin(Bθ + C) where C = arctan(b/a), a > 0, to write the model in the form A) y = (b) Find the amplitude of the oscillations of the weight. (c) Find the frequency of the oscillations of the weightarrow_forwardAn object of mass 2 grams is attached to a vertical spring with spring constant 32 grams/sec“. Neglect any friction with the air. (a) Find the differential equation y" = f(y, y') satisfied by the function y, the displacement of the object from its equilibrium position, positive downwards. Write y for y(t) and yp for y' (t). y" = Σ (b) Find r1, r2, roots of the characteristic polynomial of the equation above. r1, r2 = Σ (b) Find a set of real-valued fundamental solutions to the differential equation above. Y1 (t) = Σ Y2(t) = Σ (c) At t = 0 the object is pulled down v2 cm and the released with an initial velocity upwards of 4/2 cm/sec. Find the amplitude A > 0 and the phase shift o E (-1, 7] of the subsequent movement. A = ΣΦ Σarrow_forward
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- Consider a massless pendulum of length L and a bob of mass m at its end moving through oil. The massive bob undergoes small oscillations, but the oil regards the bob’s motion with a resistive force proportional to the speed with Fd=-b*θ. The bob is initially pulled back at t=0 with θ = αlpha with zero velocity. (a) Write down the differential equation governing the motion of the pendulum. (b) Find the angular displacement as a function of time by solving (a). Assume that b is smaller than the natural frequency (frequency in the absence of damping) of the pendulum. (c) Find the mechanical energy of the pendulum as a function of time. (d) Find the time when the mechanical energy decays to 1/e of its initial value.arrow_forwardThe motion of a particle is given by x(t) = (24cm)cos(10t) where t is in s. what is the first time at which the kinetic energy is twice the potential energy?arrow_forwardA cube, whose mass is 0.680 kg, is attached to a spring with a force constant of 122 N/m. The cube rests upon a frictionless, horizontal surface (shown in the figure below). m The cube is pulled to the right a distance A = 0.120 m from its equilibrium position (the vertical dashed line) and held motionless. The cube is then released from rest. (a) At the instant of release, what is the magnitude of the spring force (in N) acting upon the cube? N (b) At that very instant, what is the magnitude of the cube's acceleration (in m/s2)? m/s2 (c) In what direction does the acceleration vector point at the instant of release? Away from the equilibrium position (i.e., to the right in the figure). The direction is not defined (i.e., the acceleration is zero). Toward the equilibrium position (i.e., to the left in the figure). You cannot tell without more information.arrow_forward
- Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning