Problem 31 - Fundamentals of Physics

Question

A luminous point is moving at speed \(v_{o}\) toward a spherical mirror with radius of curvature \(r\), along the central axis of the mirror. Show that the image of this point is moving at speed $$ v_{I}=-\left(\frac{r}{2 p-r}\right)^{2} v_{O} $$ where \(p\) is the distance of the luminous point from the mirror at any given time. Now assume the mirror is concave, with \(r=15 \mathrm{~cm}\), and let \(v_{O}=5.0 \mathrm{~cm} / \mathrm{s}\). Find \(v_{l}\) when (b) \(p=30 \mathrm{~cm}\) (far outside the focal point), (c) \(p=8.0 \mathrm{~cm}\) (just outside the focal point), and (d) \(p=10 \mathrm{~mm}\) (very near the mirror).

Step-by-Step Solution

Answer

When \( p = 30 \text{ cm} \), \( v_I = -5.0 \text{ cm/s} \); when \( p = 8.0 \text{ cm} \), \( v_I = -1125.0 \text{ cm/s} \); and when \( p = 10 \text{ mm} \), \( v_I = -6.7 \text{ cm/s} \).

Step 1: Understand the Mirror Formula

For a spherical mirror, the mirror formula relates the object distance \( p \), the image distance \( q \), and the focal length \( f \). This is given by the mirror formula: \[ \frac{1}{f} = \frac{1}{p} + \frac{1}{q} \]Since the focal length \( f \) and the radius of curvature \( r \) are related by \( f = \frac{r}{2} \), we can substitute this relation into the formula to get: \[ \frac{2}{r} = \frac{1}{p} + \frac{1}{q} \]

Step 2: Differentiate the Mirror Formula

Differentiate both sides of the equation \( \frac{2}{r} = \frac{1}{p} + \frac{1}{q} \) with respect to time \( t \) to find the relationship between the velocities of the object and the image:\[ 0 = -\frac{1}{p^2} \frac{dp}{dt} - \frac{1}{q^2} \frac{dq}{dt} \] Rearrange to solve for \( \frac{dq}{dt} \): \[ \frac{dq}{dt} = -\frac{q^2}{p^2} \frac{dp}{dt} \] Here, \( \frac{dq}{dt} \) is the image velocity \( v_I \) and \( \frac{dp}{dt} \) is the object velocity \( v_O \), leading to: \[ v_I = -\frac{q^2}{p^2} v_O \]

Step 3: Substitute the Expression for Image Distance

Using the mirror formula \( \frac{2}{r} = \frac{1}{p} + \frac{1}{q} \), solve for \( q \) in terms of \( r \) and \( p \). After some manipulation, \( q = \frac{rp}{2p - r} \).Substitute this expression for \( q \) back into the equation for \( v_I \):\[ v_I = -\left( \frac{rp}{2p - r} \right)^2 \frac{1}{p^2} v_O \]Simplifying gives the required image speed:\[ v_I = -\left( \frac{r}{2p - r} \right)^2 v_O \]

Step 4: Calculate the Image Velocity for Different Object Distances

Use the derived formula \( v_I = -\left( \frac{r}{2p - r} \right)^2 v_O \) to calculate the image speed for different values of \( p \), with \( v_O = 5.0 \text{ cm/s} \) and \( r = 15 \text{ cm} \):(b) For \( p = 30 \text{ cm} \):\[ v_I = -\left( \frac{15}{2\cdot30 - 15} \right)^2 \times 5 = -\frac{15}{15}^2 \times 5 = -5.0 \text{ cm/s} \](c) For \( p = 8.0 \text{ cm} \):\[ v_I = -\left( \frac{15}{2\cdot8 - 15} \right)^2 \times 5 = -\left( \frac{15}{1} \right)^2 \times 5 = -1125.0 \text{ cm/s} \](d) For \( p = 10 \text{ mm} = 1.0 \text{ cm} \):\[ v_I = -\left( \frac{15}{2\cdot1 - 15} \right)^2 \times 5 = -\left( \frac{15}{-13} \right)^2 \times 5 = -6.7 \text{ cm/s} \]

Step 5: Conclusion

The image velocity \( v_I \) depends heavily on the object distance \( p \). The image velocity is inversely related to \( p \) and becomes very large as \( p \) approaches the focal length, demonstrating the sensitive dependence on distance.

Key Concepts

mirror formula

The mirror formula is fundamental in understanding how images are formed by spherical mirrors. It establishes the relationship between the object distance (\( p \)), image distance (\( q \)), and the focal length (\( f \)) of a mirror.
This formula is expressed as: \[ \frac{1}{f} = \frac{1}{p} + \frac{1}{q} \] where \( f = \frac{r}{2} \), with \( r \) being the radius of curvature of the mirror.
For spherical mirrors, this equation helps determine the position and nature of the image formed. Spheres can be of two types: concave which converges and focuses light and convex which diverges light rays. Understanding how changes in object distance affect image distance is crucial in optics.
The derivation involves simple algebraic manipulation and understanding that as \( p \) changes, \( q \) changes reciprocally due to the nature of the formula. The sensitivity of \( q \) with respect to \( p \) plays a key role in determining the velocity of the image.

image velocity

Image velocity concerns how the image of an object changes its position over time. For a spherical mirror, the velocity of the image (\( v_I \)) depends on the velocity of the object (\( v_O \)) and other factors like the distance (\( p \)) and radius of curvature (\( r \)).
The derived formula for calculating the image velocity is: \[ v_I = -\left( \frac{r}{2p - r} \right)^2 v_O \]
This formula indicates that image velocity is inversely related to the changing distance (\( p \)).
The negative sign in the formula reflects that the direction of image movement is opposite to that of the object. This insight is key when dealing with problems in real-world applications where measurements are sensitive.

• The velocity formula is especially important when objects are moving toward or away from the mirror.
• It provides insights into how the rate of motion changes due to changes in positions.

By understanding how these variables interact, one can predict motions and behavior in various configurations of optical systems.

radius of curvature

The radius of curvature (\( r \)) is a key geometrical property of spherical mirrors which directly influences their ability to focus light. It is defined as the distance between the center of the mirror sphere and its surface.
The focal length (\( f \)) of the mirror, crucial for determining the image, is directly linked to this through \( f = \frac{r}{2} \).
In practical terms, a larger radius of curvature means the mirror has a longer focal length, hence light rays are more widely spread before they are focused; a smaller radius results in rays converging sooner. For a given optical setup:

• The radius is central in determining various lens effects like magnification, inversion, and distortion.
• It impacts decisions regarding lens production and applications like telescopes or cameras.
• Understanding the radius's role aids in designing sustainable and effective optical instruments.

By observing and calculating the radius of curvature, engineers can tailor mirror properties to achieve optimal image forms for their specific needs.

concave mirror

A concave mirror is shaped such that its reflective surface curves inward, resembling a portion of the interior of a sphere. This specific shape allows it to converge light rays that strike its surface.

• Concave mirrors are often used in applications requiring focused light, such as telescopes, headlights, and mirrors in solar furnaces.
• They can produce real or virtual images depending on the object's position relative to the focal length.

In optics, concave mirrors demonstrate significant principles when analyzing image formation:
The position of an object in relation to the mirror's focal point and radius plays a vital role. If an object is placed:

• Beyond the center of curvature, the image is inverted and reduced.
• At the center of curvature, the image is inverted and of equal size.
• Between the center and focal point, the image is enlarged and inverted.
• Closer than the focal point, the image becomes virtual, upright, and magnified.

This interaction highlights the intricate balance of angles and distances that dictate image characteristics in concave mirrors, making them powerful tools for amplifying or focusing light-based information in technological and scientific explorations.