A light spring obeys Hooke's law. The spring's unstretched length is 32.0 cm. One end of the spring is attached to the top of a doorframe and a weight with mass 8.00 kg is hung from the other end. The final length of the spring is 45.0 cm.

(a)
Find its spring constant (in N/m).
(b)
The weight and the spring are taken down. Two people pull in opposite directions on the ends of the spring, each with a force of 130 N. Find the length (in m) of the spring in this situation.
(c)
What If? What would be the length (in m) of the spring if it were now attached to the vertical portion of the doorframe and stretched horizontally by a single person exerting a force of 130 N?

(a) Well, let's start with part (a). To find the spring constant, we can use Hooke's law, which states that the force exerted by a spring is directly proportional to its displacement. In equation form, it's F = -kx, where F is the force, k is the spring constant, and x is the displacement.

We're given that the spring's unstretched length is 32.0 cm, and the final length is 45.0 cm. So the displacement x will be 45.0 cm - 32.0 cm = 13.0 cm = 0.13 m.

We also know that the mass hanging from the spring is 8.00 kg. The force of gravity acting on this mass is given by F = mg, where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Since the force exerted by the spring is equal to the force of gravity acting on the mass, we have -kx = mg. Solving for k, we get k = -mg/x.

Substituting the values, we have k = -(8.00 kg * 9.8 m/s^2) / 0.13 m. Doing the math, we find that the spring constant k is approximately -6046 N/m.

Now, let's move on to part (b) and see what the spring does when two people start pulling it.

(b) Ah, the dynamic duo! When both people pull with a force of 130 N in opposite directions, the net force on the spring will be the sum of their individual forces. Since they're pulling in opposite directions, the total force will be the difference of their forces, i.e., 130 N - 130 N = 0 N.

When the net force is zero, the spring will be in equilibrium and won't stretch or compress. Therefore, its length will be the same as its original unstretched length, which is 32.0 cm or 0.32 m.

Lastly, in part (c), we're asked about attaching the spring to the vertical portion of the doorframe and stretching it horizontally.

(c) Well, in this case, we only have one person exerting a force of 130 N. Since the spring has a new orientation, it will behave differently.

To find the length of the spring, we need to know its spring constant. Luckily, we found that in part (a) to be approximately -6046 N/m.

Now, let's use Hooke's law again. Since the force F is now 130 N and we're looking for the displacement x, the equation becomes -kx = F.

Solving for x, we have x = F / k. Substituting the values, we get x = 130 N / (-6046 N/m).

Calculating it, the length of the spring in this situation is approximately -0.0215 m. Yes, you heard it right - a negative length! But don't worry, it just means that the spring is compressed in this case.

Remember, laughter is the best spring therapy, especially when dealing with negative lengths!

(a) To find the spring constant (k) of the spring, we can use Hooke's law equation:

F = -k * x

Where F is the force applied to the spring, k is the spring constant, and x is the displacement of the spring from its unstretched length.

In this case, the force applied to the spring is the weight of the hanging mass (mg), which is 8.00 kg * 9.8 m/s^2 = 78.4 N. The displacement (x) is the difference between the final length and the unstretched length, which is (45.0 - 32.0) cm = 13.0 cm = 0.13 m.

Now we can substitute the values into Hooke's law equation:

78.4 N = -k * 0.13 m

Solving for k:

k = -78.4 N / 0.13 m

k ≈ -603 N/m (Note: The negative sign indicates that the force and displacement are in opposite directions)

Therefore, the spring constant is approximately 603 N/m.

(b) To find the length (L) of the spring when two people pull in opposite directions with a force of 130 N each, we can use Hooke's law again:

F = -k * x

In this case, the force applied to the spring is the sum of the forces applied by both people, which is 130 N + 130 N = 260 N. The displacement (x) is unknown.

Now we can substitute the values into Hooke's law equation:

260 N = -603 N/m * x

Solving for x:

x = -260 N / (-603 N/m)

x ≈ 0.431 m

Therefore, the length of the spring in this situation is approximately 0.431 m.

(c) When the spring is attached to the vertical portion of the doorframe and stretched horizontally by a single person exerting a force of 130 N, the force applied to the spring and the displacement remain the same as in part (b).

So, we can use Hooke's law equation again:

260 N = -603 N/m * x

Solving for x:

x = -260 N / (-603 N/m)

x ≈ 0.431 m

Therefore, the length of the spring in this situation is also approximately 0.431 m.

(a) To find the spring constant (k), we can use Hooke's law, which states that the force exerted by a spring is directly proportional to its displacement from its equilibrium position. Mathematically, this can be expressed as:

F = -kx

Where F is the force applied to the spring, k is the spring constant, and x is the displacement of the spring from its equilibrium position.

In this case, we know that the weight attached to the spring exerts a force of F = mg, where m is the mass and g is the acceleration due to gravity. In this case, m = 8.00 kg, and g = 9.8 m/s^2.

The displacement of the spring can be calculated as the difference between the final length (45.0 cm) and the unstretched length (32.0 cm). Convert both values to meters before calculating:

Final length = 45.0 cm = 0.45 m
Unstretched length = 32.0 cm = 0.32 m

The displacement (x) = Final length - Unstretched length = 0.45 m - 0.32 m = 0.13 m

Substituting the values into Hooke's law equation, we have:

mg = kx
k = mg / x

Substituting the values, we get:

k = (8.00 kg x 9.8 m/s^2) / 0.13 m
k = 61.54 N/m

Therefore, the spring constant is 61.54 N/m.

(b) In this situation, two people are pulling in opposite directions on the ends of the spring with a force of 130 N each. Since the forces are in opposite directions, we need to subtract the forces to find the net force on the spring.

Net force = 130 N - 130 N = 0 N

Since there is no net force acting on the spring, the spring will remain in its equilibrium position, and its length will be the same as the unstretched length, which is 0.32 m.

Therefore, the length of the spring in this situation is 0.32 m.

(c) If the spring is now attached to the vertical portion of the doorframe and stretched horizontally by a single person exerting a force of 130 N, we can follow a similar approach as in part (b).

Since the person is exerting a horizontal force, it doesn't affect the vertical length of the spring. Therefore, the length of the spring will remain the same as the unstretched length, which is 0.32 m.

Therefore, the length of the spring in this situation is 0.32 m.

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