5. 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 8pounds. The entire system is placed in a medium that offers a damping force that is numerically equal to the instantaneous velocity. a. Find the equation of motion if the mass is initially released from a point 1/2 foot below the equilibrium position with a downward velocity of 1 ft/s. b. Express the equation of motion in the form x(t) = Ae sin[( )t ]

A mass weighing 4 pounds is attached to a spring whose constant is 2 Ib/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point 1 foot above the equilibrium position with a downward velocity of 14 ft/s. Determine the time (in s) at which the mass passes through the equilibrium position. (Use g = 32 ft/s? for the acceleration due to gravity.) Find the time (in s) after the mass passes through the equilibrium position at which the mass attains its extreme displacement from the equilibrium position. What is the position (in ft) of the mass at this instant? ft

2. The stiffness of a close coiled helical spring is such that it deflects 36 mm when an axial load of 10N is applied on the end hook. Calculate (i) the mass to be hung on the spring so that when set vibrating it will make one complete oscillation per second. The mass of the spring is 0.6 kg. Calculate also the maximum velocity and the maximum acceleration of the vibrating mass when initially displaced 12 mm from equilibrium.

2. A spring with spring constant 90N/m is attached to a 10kg mass with negligible friction. Determine the period that the spring mass system will oscillate for any non-zero initial conditions. T =

3: A 24 lb weight stretches a spring 6 feet. The weight hangs vertically from the spring and a damping force numerically equal to 2√√3 times the instantaneous velocity acts on the system. The weight is released from 3 feet above the equilibrium position with a downward velocity of 14 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.

7. Consider the pendulum-and-spring system depicted below. Here the mass of the pendulum rod is negligible. Derive the equation of motion and the natural frequency using the energy method. k m y you

A force of 3 N stretches a spring by 1 m.(a) Find the spring constant k.(b) Amass of 4 kg is attached to the spring.At t = 0,the mass is pulled down a distance 1 meter fromequilibrium and released with a downward velocityof 0.5 meters/second. Assuming that dampingis negligible, determine an expression for the positionof the mass at time t. Find the circular frequencyof the system and the amplitude, phase,and period of the motion.

+ b 3. A tuning fork is an example of a "resonant system", that is, one that has a low damping ratio. A model of a (half) tuning fork is rod of mass m with air drag modeled as a damper connected approximately at its middle, and a torsional spring at the base of the cantilever. Assume the cantilever rotates back and forth about its pivot by a small angle o. Because other forces are much larger, you can reasonably neglect the effect of gravity in this model. The moment of inertia of a cantilevered beam pinned at one end is equal to ml2, where I is its length. (a) What is the natural frequency of this system if m= 0.1 kg and k = 2548 Nm/rad, and 1 = 0.1 m. Please give the value in rad/s and Hz. (b) Find an upper bound on the damping coefficient b that insures that the damping ratio of the tuning fork is no greater than = 0.01. (c) For this damping ratio, what is the damped natural frequency of the system? (d) For this damping ratio, what is the decay rate (the size of the exponent…

A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position. Starting at t = 0, an external force equal to f(t) = 2sin(4t) is applied to the system. Find the equation of motion if the surrounding medium offers a damping force that is numerically equal to 8 times the instantaneous velocity. Hint: Use g = 32 ft/s2 for the acceleration due to gravity.

Hi, please answer this question. Thank you very much