Page 145 - Schaum's Outline of Differential Equations

128 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS [CHAP. 14

Supplementary Problems

14.26. A 10-lb weight is suspended from a spring and stretches it 2 in from its natural length. Find the spring constant.

14.27. A mass of 0.4 slug is hung onto a spring and stretches it 9 in from its natural length. Find the spring constant.
14.28. A mass of 0.4 g is hung onto a spring and stretches it 3 cm from its natural length. Find the spring constant.

14.29. A mass of 0.3 kg is hung onto a spring and stretches it 15 cm from its natural length. Find the spring constant.
14.30. A 20-lb weight is suspended from the end of a vertical spring having a spring constant of 40 Ib/ft and is allowed to
reach equilibrium. It is then set into motion by stretching the spring 2 in from its equilibrium position and releasing
the mass from rest. Find the position of the weight at any time t if there is no external force and no air resistance.

14.31. Solve Problem 14.30 if the weight is set in motion by compressing the spring by 2 in from its equilibrium position
and giving it an initial velocity of 2 ft/sec in the downward direction.

14.32. A 20-g mass is suspended from the end of a vertical spring having a spring constant of 2880 dynes/cm and is
allowed to reach equilibrium. It is then set into motion by stretching the spring 3 cm from its equilibrium position
and releasing the mass with an initial velocity of 10 cm/sec in the downward direction. Find the position of the mass
at any time t if there is no external force and no air resistance.
14.33. A 32-lb weight is attached to a spring, stretching it 8 ft from its natural length. The weight is started in motion by
displacing it 1 ft in the upward direction and by giving it an initial velocity of 2 ft/sec in the downward direction.
Find the subsequent motion of the weight, if the medium offers negligible resistance.

14.34. Determine (a) the circular frequency, (b) the natural frequency, and (c) the period for the vibrations described in
Problem 14.31.

14.35. Determine (a) the circular frequency, (b) the natural frequency, and (c) the period for the vibrations described in
Problem 14.32.

14.36. Determine (a) the circular frequency, (b) the natural frequency, and (c) the period for the vibrations described in
Problem 14.33.

14.37. Find the solution to Eq. (14.1) with initial conditions given by Eq. (14.2) when the vibrations are free and
undamped.

14.38. A |-slug mass is hung onto a spring, whereupon the spring is stretched 6 in from its natural length. The mass is
then started in motion from the equilibrium position with an initial velocity of 4 ft/sec in the upward direction. Find
the subsequent motion of the mass, if the force due to air resistance is —2x Ib.

14.39. A -j-slug mass is attached to a spring so that the spring is stretched 2 ft from its natural length. The mass is started
in motion with no initial velocity by displacing it yft in the upward direction. Find the subsequent motion of the
mass, if the medium offers a resistance of —4x Ib.

14.40. A -j-slug mass is attached to a spring having a spring constant of 6 Ib/ft. The mass is set into motion by displacing
it 6 in below its equilibrium position with no initial velocity. Find the subsequent motion of the mass, if the force
due to the medium is —4x Ib.
14.41. A y-kg mass is attached to a spring having a spring constant of 8 N/m. The mass is set into motion by displacing
it 10 cm above its equilibrium position with an initial velocity of 2 m/sec in the upward direction. Find the
subsequent motion of the mass if the surrounding medium offers a resistance of -4iN.

14.42. Solve Problem 14.41 if instead the spring constant is 8.01 N/m.
14.43. Solve Problem 14.41 if instead the spring constant is 7.99 N/m.