-An object whose height is 5.8 cm is at a distance of 9.5 cm from a spherical concave mirror. Its image is real and has a height of 9.2 cm. Calculate the radius of curvature of the mirror.
I got 0.1165 m for my answer.
-How far from the mirror is it necessary to place the above object in order to have a virtual image with a height of 9.2 cm?
I know that I have two unknowns to solve for, but how do I go about solving them?
Use the desired image height and the object height to get the magnification ratio: it is 9.2/5.8 = 1.586
Since you want the image to be virtual, the Do value must be negative, and such that |Di/Do| = 1.58 (the magnification)
Therefore Di = -1.58 Do
Use that together with the focusing equation
1/Do + 1/Di = 2/R
to solve for Do. You already know R.
I get as far as 1/Do + 1/-1.58 = 17.167, but I'm not sure what my next step is.
Wrong equation
1/Do + 1/-1.58Do = 17.167
Solve for Do
To calculate the radius of curvature of the mirror in the first question, you can use the mirror formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the mirror,
- v is the image distance from the mirror (positive for real images, negative for virtual images),
- u is the object distance from the mirror (positive for objects on the same side as the incident light, negative for objects on the opposite side).
In this case, the object distance (u) is given as 9.5 cm and the image distance (v) is the negative of the height of the image (9.2 cm). Plugging these values into the formula, we get:
1/f = 1/-9.2 - 1/9.5
Simplifying this expression will give you the reciprocal of the focal length. To find the radius of curvature (R), you can use the equation:
R = 2f
Once you have the reciprocal of the focal length, you can find the radius of curvature by taking its reciprocal and multiplying by 2.
Now, moving on to the second question, if you want to find the distance from the mirror at which the object should be placed to obtain a virtual image with a certain height, you can use the magnification formula:
magnification (m) = -v/u
Here, the magnification is given as the ratio of the image height (9.2 cm) to the object height (5.8 cm). Solving this equation for v, we get:
v = -m × u
Since the magnification is negative for virtual images, you can plug in the values for magnification (m = 9.2/5.8) and object distance (u = 9.5 cm) to find the image distance from the mirror (v). The resulting value of v will give you the distance from the mirror at which the object should be placed.
Remember to keep track of the positive/negative signs when plugging in the values and solving the equations.