Example 7  An Object on a Horizontal Spring

Figure 10.19 shows an object of mass m=0.200 kg that is vibrating on a horizontal frictionless table. The spring has a spring constant k=545 N/m. It is stretched initially to x0= 4.50 cm and then released from rest (see part A of the drawing). Determine the final translational speed vf of the object when the final displacement of the spring is (a) xf=2.25 cm and (b) xf=0 cm.

The total mechanical energy of this system is entirely elastic potential energy (A), partly elastic potential energy and partly kinetic energy (B), and entirely kinetic energy (C).
Figure 10.19  The total mechanical energy of this system is entirely elastic potential energy (A), partly elastic potential energy and partly kinetic energy (B), and entirely kinetic energy (C).

Reasoning  The conservation of mechanical energy indicates that, in the absence of friction (a nonconservative force), the final and initial total mechanical energies are the same (see Figure 10.18):

Since the object is moving on a horizontal table, the final and initial heights are the same: hf=h0. The object is not rotating, so its angular speed is zero: wf=w0=0 rad/s. And, as the problem states, the initial translational speed of the object is zero, v0=0 m/s. With these substitutions, the conservation-of-energy equation becomes
from which we can obtain vf:

Solution

(a) Since x0=0.0450 m and xf=0.0225 m, the final translational speed is
The total mechanical energy at this point is partly translational kinetic energy () and partly elastic potential energy (). The total mechanical energy E is the sum of these two energies: E=0.414 J +0.138 J=0.552 J. Because the total mechanical energy remains constant during the motion, this value equals the initial total mechanical energy when the object is stationary and the energy is entirely elastic potential energy ().
(b) When x0=0.0450 m and xf=0 m, we have
Now the total mechanical energy is due entirely to the translational kinetic energy , since the elastic potential energy is zero. Note that the total mechanical energy is the same as it is in part (a). In the absence of friction, the simple harmonic motion of a spring converts the different types of energy between one form and another, the total always remaining the same.



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