After a mass weighing 10 pounds is attached to a 5-foot spring, the spring measures 7 feet. This mass is removed and replaced with another mass that weighs 8 pounds. The entire system is placed in a medium that offers a damping force that is numerically equal to the instanta- neous velocity. (a) Find the equation of motion if the mass is initially released from a point foot below the equilibrium position with a downward velocity of 1 ft/s. (b) Express the equation of motion in the form given in (23). (c) Find the times at which the mass passes through the equilibrium position heading downward.
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- A force of 5 pounds stretches a spring 1 foot. A mass weighing 6.4 pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 1.2 times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 foot above the equilibrium position. x(t) = (b) Express the equation of motion in the form x(t) x(t) ft = ft = Ae¯^t sin(√√ ² - ^²t + 4) 9), which is given in (23) of Section 3.8. (Round p to two decimal places.) (c) Find the first time at which the mass passes through the equilibrium position heading upward. (Round your answer to three decimal places.) SA 140 lb weight stretches a spring 14 feet. The weight hangs vertically from the spring and a damping force numerically equal to 5√√7 times the instantaneous velocity acts on the system. The weight is released from 7 feet above the equilibrium position with a downward velocity of 30 ft/s. (a) Determine the time (in seconds) at which the mass passes through the equilibrium position. (b) Find the time (in seconds) at which the mass attains its extreme displacement from the equilibrium position. Round your answer to 4 decimals. Round your answer to 4 decimals.A simple pendulum with a length of 2.13 m and a mass of 6.94 kg is given an initial speed of 2.66 m/s at its equilibrium position. (a) Assuming it undergoes simple harmonic motion, determine its period. (b) Determine its total energy. (c) Determine its maximum angular displacement. (For large v, and/or small /, the small angle approximation may not be good enough here.)
- The period of a simple harmonic oscillator is 6.02 s. At time t = 0 s, the position of the oscillator is x(0) = A. Calculate the first time, in seconds, when the kinetic energy is 5.74 times smaller than the potential energy.A 16 lb weight is suspended from a spring having spring constant 5 lb/ft. Assume that an external force given by 24 sin (10t) and a damping force with damping constant 4, are acting on the spring. Initially the weight is at rest at its equilibrium position. Find the position of the weight at any time. Find the steady state solution. Find the amplitude, period and frequency of the steady state solution. Determine the velocity of the weight at any timeThe support of a 130 lb mass is moved harmonically with an amplitude of 0.15-inch at 6 Hz. It hangs from a spring with a stiffness of 75 lb/ft. The damping ratio is 0.2. Calculate the amplitude of the mass in inches and the phase angle.
- A spring with a mass of 2 kg has natural length 0.5 m. A force of 25.6 N is required to maintain it stretched to a length of 0.7 m. Suppose that the spring is immersed in a fluid with damping constant c=40. (a) Using Newton's Second and Hooke's Law, derive the 2nd order differential equation for the following mass-spring system. equilibrium position Figure 3 for Q3. (b) If the spring starts from equilibrium and is stretched to a length of 0.7 m and then released with initial velocity 0.6 m/s, find the position of the mass at any time t.A force of 9 pounds stretches a spring 1 foot. A mass weighing 6.4 pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 1.2 times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 foot above the equilibrium position. ft x(t) = = -λt (b) Express the equation of motion in the form x(t) = Ae¯ sin(√²-2²t+ x(t) = = ft q²t + 9), which is given in (23) of Section 3.8. (Round p to two decimal places.) (c) Find the first time at which the mass passes through the equilibrium position heading upward. (Round your answer to three decimal places.) SAn object stretches a spring 6 centimeters in equilibrium. Find its displacement for t>0 if it is initially displaced 3 centimeters above equilibrium and given a downward velocity of 6 centimeters/sec. Assume that there’s no damping.
- On a nice spring day you find yourself relaxing under an old oak tree. You pull the end of one of the tree's branches down (negative maximum displacement) about 2 ft and let it go (t = 0 s). You note that the resulting motion appears to be from a simple harmonic oscillator with a period of about 2.5 seconds. It takes the branch about 8 oscillations to decrease the amplitude by a factor of 2. (a) What evidence could you have to claim the branch undergoes simple harmonic motion? (be brief) (b) Describe the energy transfers in the branch + earth system during an entire cycle of the motion. Focus on the three most important types of energy in the system. Feel free to use a plot or diagram to illustrate how/ when the energy changes forms. (c) What is the time constant τ for the amplitude as a function of time during the damping of the oscillation.A 0.250-kg glider on a horizontal air track is attached to an ideal spring. When the glider is pulled 0.400 m from its equilibrium position and released from rest, it oscillates with an angular frequency of 11.0 rad/s. (Ignore friction.) (a) Find the force constant of the spring. (b) Find the period of the oscillations. (c) Find the maximum speed of the mass.A mass is attached to two springs as shown below. Both springs have a k = 100 N/m and the mass is 0.8 kg. m 2T√ (a) Show that the period is given by T = (assume no damping). (b) Now it is observed that the amplitude drops by 5% after each oscillation. What is the value of the dampening constant b (you may assume it is lightly damped)? ! ееее m 2k m шеее