17. Periodic Motion

(I) A 1.28-kg mass oscillates according to the equation 𝓍 = 0.650 cos7.40 t where 𝓍 is in meters and t in seconds. Determine

(d) the kinetic energy and potential energy when 𝓍 = 0.260 m.

(I) If one oscillation has 3.0 times the energy of a second one of equal frequency and mass, what is the ratio of their amplitudes?

(II) (b) What fraction of the total energy of a SHO is kinetic and what fraction potential when the displacement is one third the amplitude?

(II) A mass resting on a horizontal, frictionless surface is attached to one end of a spring; the other end of the spring is fixed to a wall. It takes 3.2 J of work to compress the spring by 0.13 m. The mass is then released from rest and experiences a maximum acceleration of 12m/s² . Find the value of

(b) the mass.

(II) A mass resting on a horizontal, frictionless surface is attached to one end of a spring; the other end of the spring is fixed to a wall. It takes 3.2 J of work to compress the spring by 0.13 m. The mass is then released from rest and experiences a maximum acceleration of 12m/s² . Find the value of

(a) the spring constant .

(II) An object with mass 2.7 kg is executing simple harmonic motion, attached to a spring with spring constant k = 310 N/m. When the object is 0.020 m from its equilibrium position, it is moving with a speed of 0.60 m/s. n.

(b) Calculate the maximum speed attained by the object.

A 280-kg wooden raft floats on a lake. When a 68-kg man stands on the raft, it sinks 3.5 cm deeper into the water. When he steps off, the raft oscillates for a while.

(b) What is the total energy of oscillation (ignoring damping)?

A diving board oscillates with simple harmonic motion of frequency 3.0 cycles per second. What is the maximum amplitude with which the end of the board can oscillate in order that a pebble placed there (Fig. 14–42) does not lose contact with the board during the oscillation?

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(II) Draw a graph like Fig. 14–11 for a horizontal spring whose spring constant is 95 N/m and which has a mass of 75 g on the end of it. Assume the spring was started with an initial amplitude of 2.0 cm. Neglect the mass of the spring and any friction with the horizontal surface. Use your graph to estimate

(a) the potential energy for 𝓍 = 1.5 cm.

(II) Draw a graph like Fig. 14–11 for a horizontal spring whose spring constant is 95 N/m and which has a mass of 75 g on the end of it. Assume the spring was started with an initial amplitude of 2.0 cm. Neglect the mass of the spring and any friction with the horizontal surface. Use your graph to estimate ,

(b) the kinetic energy, for 𝓍 = 1.5 cm.

(II) Draw a graph like Fig. 14–11 for a horizontal spring whose spring constant is 95 N/m and which has a mass of 75 g on the end of it. Assume the spring was started with an initial amplitude of 2.0 cm. Neglect the mass of the spring and any friction with the horizontal surface. Use your graph to estimate

(c) the speed of the mass, for 𝓍 = 1.5 cm.