a 4.0 cm high object is placed 5.10 cm from a concave mirror whose redius of curvature is 21.2 cm. Find the inmage distance and the image size.
To find the image distance and image size, you can use the mirror equation and magnification formula. Here's how you can calculate them step by step:
1. Identify the given values:
- Object height (h₀) = 4.0 cm
- Object distance (d₀) = 5.10 cm
- Radius of curvature (R) = 21.2 cm
2. Determine the focal length (f) using the formula: f = R/2
- For concave mirrors, the focal length is positive.
- Substitute the given values: f = 21.2 cm / 2 = 10.6 cm
3. Use the mirror equation to find the image distance (dᵢ):
- The mirror equation is: 1/f = 1/d₀ + 1/dᵢ
- Rearrange the equation to isolate dᵢ: 1/dᵢ = 1/f - 1/d₀
- Substitute the values: 1/dᵢ = 1/10.6 - 1/5.10
- Evaluate: 1/dᵢ = 0.0943 - 0.1961 = -0.1018
- Take the reciprocal to find dᵢ: dᵢ = 1 / (-0.1018) ≈ -9.824 cm
(Note: the negative sign indicates a virtual image formed on the same side as the object.)
4. Calculate the magnification (M) using the formula: M = -dᵢ / d₀
- Substitute the values: M = -(-9.824 cm) / 5.10 cm ≈ 1.92
5. Find the image size (hᵢ) using the magnification formula: M = hᵢ / h₀
- Rearrange the equation to isolate hᵢ: hᵢ = M * h₀
- Substitute the values: hᵢ ≈ 1.92 * 4.0 cm ≈ 7.68 cm
Therefore, the image distance is approximately -9.824 cm (virtual image), and the image size is about 7.68 cm.