A single mass m 1 = 4 kg hangs from a spring in a motionless elevator. The sprin

Question

A single mass m1 = 4 kg hangs from a spring in a motionless elevator. The spring is extended x = 12 cm from its unstretched length.

1) What is the spring constant of the spring?

2) Now, three masses m1 = 4 kg, m2 = 12 kg and m3 = 8 kg hang from three identical springs in a motionless elevator. The springs all have the same spring constant that you just calculated above.

What is the force the top spring exerts on the top mass?

3) What is the distance the lower spring is stretched from its equilibrium length?

4) Now the elevator is moving downward with a velocity of v = -2.3 m/s but accelerating upward with an acceleration of a = 5.5 m/s2. (Note: an upward acceleration when the elevator is moving down means the elevator is slowing down.)

What is the force the bottom spring exerts on the bottom mass?

5) What is the distance the upper spring is extended from its unstretched length?

6) What is the distance the MIDDLE spring is extended from its unstretched length?

Explanation / Answer

1.What is the spring constant of the spring? [4 kg, 12cm extension]

NOTE: Although the question says "x = 12 cm" (implying a horizontal extension using the usual coordinate system), the word "hanging" implies vertical, or the y-direction.

(1) F = k * x is Hooke's Law. Applied to this situation, where the force is gravity:

(2) Fg = m * g = k * x ==>>

(3) k = m1 * g / x [in N / m]

= 4 * 9.8 / 0.12

= 326.66 N/m <<<=== Answer to 1.

2.Now, three masses m1 = 4 kg, m2 = 12 kg and m3 = 8 kg hang from three identical springs in a motionless elevator. The springs all have the same spring constant that you just calculated above.
What is the force the top spring exerts on the top mass?

Since the top spring s1 is supporting all 3 masses, it is exerting a force equal to gravity's downward pull but in the opposite direction:

(4) Fg = (m1 + m2 + m3) * g

= 24 * 9.81

= ~235.2.N <<<===Answer to 2.


3. What is the distance the lower spring is stretched from its equilibrium length?

The lower spring s3 is supporting only m3; applying Hooke's law:

(5) x = Fg / k = m * g / k

Notice, however that m3 = 2 * m1, therefore:

(6) x3 = 2 * x1 = 2 * 12 = 24cm <<<=== Answer to 3.


4.Now the elevator is moving downward with a velocity of v = -2.3 m/s but accelerating upward with an acceleration of a = 5.5 m/s2. (Note: an upward acceleration when the elevator is moving down means the elevator is slowing down.)
What is the force the bottom spring exerts on the bottom mass?

The downward velocity has no effect on the force situation, it is only changes in velocity (plus, of course, gravity, which is always there) that require a force. At constant velocity, the bottom spring s3 is supporting its mass m3 to balance gravity.

As the elevator slows, though, it also ends up slowing down the spring arrangement, too. However, because the stretching takes time, it means that some damped harmonic motion will be set up in the spring chain.

When the motion has finally damped out, the net force the bottom spring s3 exerts on m3 has two components--that of gravity and of the deceleration of the elevator:

(7) F3net = m3 * (g + a)

= 8 * (9.8 + 5.5)

= ~ 122.4N Ans. of 4


5.What is the distance the upper spring is extended from its unstretched length?

Assuming this question applies to the decelerating elevator (to make the question different from #2.), the upper spring s1 is not only balancing the force of gravity on all 3 masses but also counteracting the decelerating force:

(8) F = (m1 + m2 + m3) * (g + a)

= ~235.2N [gravity, from (4) ] + 24 *5.5

= ~235.2. + ~132

= 367.2N

Again, using Hooke's law:

x = 367.2 / 326.66

= 1.12m = 112cm


6.What is the distance the MIDDLE spring is extended from its unstretched length?

Assuming again that this question applies to the decelerating elevator, the middle spring s2 is not only balancing the force of gravity on m2 and m3 but also counteracting the decelerating force:

F = (m2 + m3) * (g + a)

= (12 + 8) * (9.81 + 5.5)

x = F / k

= (12 + 8) * (9.81 + 5.5) / 326.67

= 0.936m

= 93.6cm Ans.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.