A cylindrical glass is 6cm wide. When you look from above at an angle you see the edge of the bottom of the glass. When the glass is then filled with water (1.33) you see the center of the bottom of the glass. How tall is the glass?
To find the height of the glass, we can use the principle of refraction.
Step 1: Let's assume the height of the glass is 'h' cm.
Step 2: When you look from above at an angle, you see the edge of the bottom of the glass. This means that the light rays coming from the edge of the bottom get refracted, and the apparent position of the bottom moves down.
Step 3: Since the glass is cylindrical, the apparent position of the bottom will be at the same height as the top of the glass. Therefore, the apparent height from above is equal to the height of the cylinder (h).
Step 4: When the glass is filled with water (refractive index of water = 1.33), you see the center of the bottom of the glass. This means that the light rays coming from the center of the bottom do not get refracted and appear to be coming straight from the bottom.
Step 5: The apparent position of the bottom when the glass is filled with water will be at the same height as the original bottom (before filling). Therefore, the apparent height when filled with water is also h.
Step 6: According to the principle of refraction, the apparent height when filled with water is the real height (h) divided by the refractive index of water (1.33).
So, h/1.33 = h.
Step 7: To find the value of h, we can solve the equation:
h/1.33 = h.
Step 8: Multiply both sides of the equation by 1.33 to isolate h:
1.33 * h/1.33 = h * 1.33.
This simplifies to:
h = 1.33h.
Step 9: Divide both sides of the equation by h:
h/h = 1.33h/h.
This simplifies to:
1 = 1.33.
Step 10: Since 1 does not equal 1.33, this means that the equation has no valid solution.
Therefore, there is no height that satisfies the given conditions.
To find the height of the glass, we need to use the concept of refraction and Snell's law.
Let's break down the problem step by step:
Step 1: Identify the initial situation:
In the initial situation, when you look from above at an angle, you see the edge of the bottom of the glass. This implies that light rays coming from the edge of the bottom of the glass bend at the air-water interface, causing the apparent position to shift.
Step 2: Identify the final situation:
In the second situation, when the glass is filled with water (refractive index = 1.33), you see the center of the bottom of the glass. This indicates that the light rays coming from the center of the bottom of the glass do not bend much or undergo minimal refraction.
Step 3: Determine the incident and refracted angles:
In this scenario, the incident ray is the light ray coming from the edge of the bottom of the glass, and the refracted ray is the light ray coming from the center of the bottom of the glass.
Step 4: Apply Snell's law:
Snell's law relates the incident and refracted angles with the refractive indices of the two mediums (in this case, air and water). It is represented as:
n1 * sin(theta1) = n2 * sin(theta2)
where
n1 = refractive index of the initial medium (air)
theta1 = incident angle
n2 = refractive index of the final medium (water)
theta2 = refracted angle
Step 5: Calculate the incident and refracted angles:
Since we are looking from above, and considering a cylindrical glass, the incident and refracted angles can be assumed to be small.
Therefore, sin(theta1) ≈ tan(theta1) and sin(theta2) ≈ tan(theta2).
Step 6: Assign variables and solve the equations:
Let's assign variables for the measurements:
- r = radius of the glass (6cm)
- h = height of the glass (unknown)
Now, we can rewrite Snell's law as:
tan(theta1) = (r / h)
tan(theta2) = (r / h)
Since tan(theta1) = tan(theta2), we can equate the two equations:
(r / h) = (r / h)
Step 7: Solve for height (h):
Cross-multiplying the equation (r / h) = (r / h), we get:
r * h = r * h
This equation indicates that the radius of the glass is equal to the height of the glass. Therefore, using the given radius of 6 cm, we can conclude that the height of the glass is also 6 cm.
So, the height of the glass is 6 cm.