Question

Problem Statement: Suppose that a body of mass m hangs on one end of a spring, and the other end is fastened to the ceiling. Measure the body's vertical position along an x-axis. Place x = 0 at the equilibrium position (where the spring is neither stretched nor compressed). Say x > 0 indicates a stretched spring and x < 0 indicates a compressed spring. equilibrium position According to Hooke's law and Newton's Second Law, the position x(t) of the body at time t must satisfy the differential equation: mx'' + cx' + kx = F(t) where k is a positive constant called the spring constant, c is the positive damping constant, and F(t) is an external forcing function. You will investigate a variety of situations and analyze the effect each case has on the behavior of the system. Assume the body has mass m = 1 kg and the spring constant is k = 4 N/m. Also suppose x is measured in meters and t in seconds. 2. Damped Free Vibrations You will now investigate the effect of damping on the system in the absence of an external force. Assume again that the object is initially stretched 0.1 meters beyond its equilibrium position with an initial velocity of 0.4 m/s. CASE 2: Overdamped (c^2 > 4mk) Let c = 5 N·s/m (a) Determine the position of the object at time t. (b) Graph x(t) (a hand-drawn sketch is fine). (c) Describe the motion of the object. In particular, what happens to the amplitude of the oscillations as t ? ??

          Problem Statement: Suppose that a body of mass m hangs on one end of a spring,
and the other end is fastened to the ceiling. Measure the body's vertical position
along an x-axis. Place x = 0 at the equilibrium position (where the spring is neither
stretched nor compressed). Say x > 0 indicates a stretched spring and x < 0
indicates a compressed spring.

equilibrium position

According to Hooke's law and Newton's Second Law, the position x(t) of the body
at time t must satisfy the differential equation:

mx'' + cx' + kx = F(t)

where k is a positive constant called the spring constant, c is the positive damping
constant, and F(t) is an external forcing function.

You will investigate a variety of situations and analyze the effect each case has on
the behavior of the system. Assume the body has mass m = 1 kg and the spring
constant is k = 4 N/m. Also suppose x is measured in meters and t in seconds.

2. Damped Free Vibrations
You will now investigate the effect of damping on the system in the absence
of an external force. Assume again that the object is initially stretched 0.1
meters beyond its equilibrium position with an initial velocity of 0.4 m/s.

CASE 2: Overdamped (c^2 > 4mk)

Let c = 5 N·s/m

(a) Determine the position of the object at time t.

(b) Graph x(t) (a hand-drawn sketch is fine).

(c) Describe the motion of the object. In particular, what happens to the
amplitude of the oscillations as t ? ??

Problem Statement: Suppose that a body of mass m hangs on one end of a spring,
and the other end is fastened to the ceiling. Measure the body's vertical position
along an x-axis. Place x = 0 at the equilibrium position (where the spring is neither
stretched nor compressed). Say x > 0 indicates a stretched spring and x < 0
indicates a compressed spring.

equilibrium position

According to Hooke's law and Newton's Second Law, the position x(t) of the body
at time t must satisfy the differential equation:

mx” + cx' + kx = F(t)

where k is a positive constant called the spring constant, c is the positive damping
constant, and F(t) is an external forcing function.

You will investigate a variety of situations and analyze the effect each case has on
the behavior of the system. Assume the body has mass m = 1 kg and the spring
constant is k = 4 N/m. Also suppose x is measured in meters and t in seconds.

2. Damped Free Vibrations
You will now investigate the effect of damping on the system in the absence
of an external force. Assume again that the object is initially stretched 0.1
meters beyond its equilibrium position with an initial velocity of 0.4 m/s.

CASE 2: Overdamped (c^2 > 4mk)

Let c = 5 N·s/m

(a) Determine the position of the object at time t.

(b) Graph x(t) (a hand-drawn sketch is fine).

(c) Describe the motion of the object. In particular, what happens to the
amplitude of the oscillations as t ? ??

Added by Ana C.

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