A certain coiled wire with uneven windings has the property that to stretch it an amount s from its relaxed length requires a force that is given by , so its behavior is different from a normal spring. You suspend this device vertically, and its unstretched length is 19.0 cm. You hang a mass of 17.0 grams from the device, and you observe that the length is now 21.0 cm. What is b? (Start your analysis from the Momentum Principle.)

You hold the 17.0 gram mass and throw it downward, releasing it when the length of the spring-like device is 24.0 cm and the speed of the mass is 5.40 m/s. One millisecond later, what is the approximate change in the speed of the mass?

At this same time, one millisecond after throwing, what is the approximate change in the stretch of the device?

To solve these problems, we can start by applying the momentum principle. The momentum principle states that the net external force acting on an object is equal to the rate of change of its momentum.

1. Solve for b using the momentum principle:
First, let's calculate the change in length of the device when the mass is hung from it.
∆s = 21.0 cm - 19.0 cm
= 2.0 cm

Next, convert the mass to kg:
mass = 17.0 g = 0.017 kg

Applying the momentum principle:
F_net = m*(∆v/∆t)
b*s = m*(∆v/∆t)

Simplifying the equation, we can relate the force to b and s:
b = (m*(∆v/∆t))/s

Since the problem does not provide the value of ∆v/∆t, we cannot determine the exact value of b without further information.

2. Approximate change in the speed of the mass:
One millisecond after throwing, the approximate change in the speed of the mass can be obtained by finding the net force acting on the mass and using Newton's second law: F_net = m*a

Given:
Unstretched length (lo) = 19.0 cm
Stretched length (lv) = 24.0 cm
Velocity (v) = 5.40 m/s

Calculate the change in length:
∆s = lv - lo
= 24.0 cm - 19.0 cm
= 5.0 cm

Convert the change in length to meters:
∆s = 5.0 cm * (1 m/100 cm)
= 0.05 m

Assuming the device is stretched uniformly, we can estimate the acceleration (a) using the equation for the spring force:
F_spring = k*∆s

Rearranging the equation and substituting the known values:
k = F_spring / ∆s

To find F_spring, we can use the relationship between force and momentum:
F_net = m*(∆v/∆t)

Substituting the known values:
F_net = 0.017 kg * (∆v/∆t)

Now, we can solve for the approximate change in velocity (∆v):
∆v ≈ F_net * ∆t / m

Given that ∆t = 1 ms = 0.001 s:
∆v ≈ (0.017 kg * (∆v/∆t)) * 0.001 s / 0.017 kg

Simplifying, we find:
∆v ≈ (∆v/∆t) * 0.001 s

Since we do not have the exact values of (∆v/∆t), we cannot calculate the approximate change in the speed of the mass without further information.

3. Approximate change in the stretch of the device:
One millisecond after throwing, the approximate change in the stretch of the device can be determined by considering the change in speed of the mass.

Since the speed of the mass changes, the force applied to the device will also change. This will cause a change in the stretch of the device. However, without the exact values of the force or spring constant, we cannot determine the approximate change in the stretch of the device accurately.