Two systems are formed from a converging lens and a diverging lens, as shown in parts a and b of the drawing. (The point labeled "Fconverging" is the focal point of the converging lens.) An object is placed inside the focal point of lens 1 at a distance of 7.70 cm to the left of lens 1. The focal lengths of the converging and diverging lenses are 15.0 and -20.0 cm respectively. The distance between the lenses is 50.0 cm. Determine the final image distance for each system, measured with respect to lens 2.

To determine the final image distance for each system, we need to apply the lens formula to each lens separately and then combine the results.

Let's start with the converging lens (lens 1).

Given:
Object distance (u1) = -7.70 cm (negative because it is to the left of the lens)
Focal length (f1) = 15.0 cm

Using the lens formula: 1/f1 = 1/v1 - 1/u1

We can rearrange the formula to solve for the image distance (v1):
1/v1 = 1/f1 - 1/u1
1/v1 = 1/15.0 - 1/(-7.70)

Calculating the right-hand side:
1/v1 = 0.06667 + 0.12987
1/v1 = 0.19654

Now, we can solve for v1:
v1 = 1/0.19654
v1 = 5.08 cm (rounded to two decimal places)

Next, let's move on to the diverging lens (lens 2).

Given:
Object distance (u2) = 50.0 cm (distance between the lenses)
Focal length (f2) = -20.0 cm

Using the lens formula: 1/f2 = 1/v2 - 1/u2

We can rearrange the formula to solve for the image distance (v2):
1/v2 = 1/f2 - 1/u2
1/v2 = 1/-20.0 - 1/50.0

Calculating the right-hand side:
1/v2 = -0.05 - 0.02
1/v2 = -0.07

Now, we can solve for v2:
v2 = 1/-0.07
v2 = -14.29 cm (rounded to two decimal places)

Since we are measuring the final image distance with respect to lens 2, we need to find the distance between the lenses (d2) by subtracting the image distance (v1) from the object distance (u2):
d2 = u2 - v1
d2 = 50.0 - 5.08
d2 = 44.92 cm (rounded to two decimal places)

Therefore, the final image distance for each system, measured with respect to lens 2, is approximately 44.92 cm.

To determine the final image distance for each system, we can use the lens formula:

1/f = 1/v - 1/u

where:
f is the focal length of the lens
v is the image distance from the lens
u is the object distance from the lens (measured along the principal axis)

For System 1 (converging lens), the object is placed inside the focal point of lens 1 at a distance of 7.70 cm to the left of lens 1. The focal length of the converging lens is +15.0 cm.

Since the object is inside the focal point, the object distance (u) is negative: u = -7.70 cm.
To determine the image distance (v) after lens 1, we can use the lens formula:

1/f1 = 1/v1 - 1/u1

Substituting the known values:

1/15 = 1/v1 - 1/-7.70

Simplifying,

1/15 = 1/v1 + 1/7.70

To get v1, we need to solve for it. Rearranging the equation,

1/v1 = 1/15 - 1/7.70

1/v1 = (7.70 - 15) / (15 * 7.70)

v1 = (15 * 7.70) / (15 - 7.70)

v1 = 118.5 / 7.30

v1 ≈ 16.23 cm

Now, for System 2 (diverging lens), the image formed by lens 1 acts as the object for lens 2. The image distance (u2) becomes the object distance for lens 2, and the focal length of lens 2 is -20.0 cm.

The object distance (u2) for lens 2 is given by u2 = v1.

To determine the image distance (v2) after lens 2, we can again use the lens formula:

1/f2 = 1/v2 - 1/u2

Substituting the known values:

1/-20 = 1/v2 - 1/16.23

Simplifying,

1/-20 = 1/v2 - 1/16.23

To get v2, we need to solve for it. Rearranging the equation,

1/v2 = 1/-20 + 1/16.23

1/v2 = (-16.23 - 20) / (-20 * 16.23)

v2 = (-20 * 16.23) / (-20 - 16.23)

v2 = -324.6 / -36.23

v2 ≈ 8.95 cm

Therefore, the final image distance for System 1 (measured with respect to lens 2) is approximately 16.23 cm, and for System 2 it is approximately 8.95 cm.

System A: The final image distance for system A is -25.0 cm, measured with respect to lens 2.

System B: The final image distance for system B is -35.0 cm, measured with respect to lens 2.