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A 1.00-kg object slides to the right on a surface having a coefficient of kinetic friction 0.250 (Fig. P7.68a). The object has a speed of vi = 3.00 m/s when it makes contact with a light spring (Fig. P7.68b) that has a force constant of 50.0 N/m. The object comes to rest after the spring has been compressed a distance d (Fig. P7.68c). The object is then forced toward the left by the spring (Fig. P7.68d) and continues to move in that direction beyond the spring’s unstretched position. Finally, the object comes to rest a distance D to the left of the unstretched spring (Fig. P7.68e). Find (a) the distance of compression d, (b) the speed v at the unstretched position when the object is moving to the left (Fig. P7.68d), and (c) the distance D where the object comes to rest.
Figure P7.68
(a)

Distance of compression.
Answer to Problem 68P
The distance of compression is
Explanation of Solution
Write the energy conservation equation between second and third picture.
Here
Write the equation for change in kinetic energy,
Here
Write the equation for change in potential energy,
Here
The final kinetic energy zero as the final velocity zero. The initial elastic potential energy is also zero as the spring is not extended initially.
Write the equation for change in internal energy
Here
Substitute (II), (III) and (IV) in (I)
Conclusion:
Substitute
Then,
The distance of compression is
(b)

Speed at the un stretched position when the object is moving left.
Answer to Problem 68P
The speed is
Explanation of Solution
Write the energy conservation equation between picture two and four
Substitute (II) and (IV) in (VI)
Substitute
Rewrite (VIII) for
Conclusion:
Substitute
The speed is
(c)

The distance at which the object comes to rest.
Answer to Problem 68P
The distance is
Explanation of Solution
Consider the motion from picture two to five
Substitute
Rearrange (X) in terms of
Conclusion:
Substitute
The distance is
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Chapter 7 Solutions
Principles of Physics: A Calculus-Based Text, Hybrid (with Enhanced WebAssign Printed Access Card)
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