Problem 113 - Physics

Question

At age forty, a certain man requires contact lenses \((f=65.0 \mathrm{~cm})\) to read a book held \(25.0 \mathrm{~cm}\) from his eyes. At age forty-five, while wearing these contacts he must now hold a book \(29.0 \mathrm{~cm}\) from his eyes. (a) By what distance has his near point changed? (b) What focal length lenses does he require at age forty-five to read a book at \(25.0 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

His near point has changed by 4.0 cm, and he needs lenses with a focal length of approximately 181.16 cm.

Step 1: Understanding the Problem

A man initially uses contact lenses with focal length \( f = 65.0 \text{ cm} \) for reading at \( 25.0 \text{ cm} \). At age forty-five, he can only read at \( 29.0 \text{ cm} \). We need to find how much his near point has changed and the new lens focal length he needs to read at \( 25.0 \text{ cm} \).

Step 2: Calculating Near Point Change

Initially, his far point with the contacts brought his near point to \( 25.0\text{ cm} \). At forty-five, with the same contacts, his near point moves to \( 29.0\text{ cm} \). Thus, the change in near point is \( 29.0 - 25.0 = 4.0 \text{ cm} \).

Step 3: Determining New Focal Length Requirement

To find the focal length he requires now, we use the lens formula. The lens formula is \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \). Here \( v = 25.0 \text{ cm} \), and \( u = 29.0 \text{ cm} \). Calculate \( \frac{1}{f} = \frac{1}{25.0} - \frac{1}{29.0} \).

Step 4: Solving for New Focal Length

Calculate \( \frac{1}{25.0} = 0.04 \) and \( \frac{1}{29.0} \approx 0.03448 \). Subtract these to find \( \frac{1}{f} \approx 0.04 - 0.03448 = 0.00552 \). Therefore, \( f \approx \frac{1}{0.00552} \approx 181.16 \text{ cm} \).

Key Concepts

Focal Length

In optical physics, the focal length of a lens is a critical attribute that determines how the lens converges or diverges light. The focal length is the distance from the center of the lens to the point where the light rays converge to form a sharp image. It is usually measured in centimeters or meters.

• Lenses with a short focal length bend light more sharply and bring it to focus in a shorter distance. This makes them suitable for reading glasses or close-up work.
• Lenses with a long focal length bend light less sharply, forming a focus further away. These lenses are typically used for distance viewing.

Understanding focal length helps in choosing the right lens for activities, like reading which typically requires a focus at close distances, often around 25 cm from the eyes. For someone whose eyesight changes, as in our exercise, determining the correct focal length involves understanding the specific needs of your vision change.

Lens Formula

The lens formula is an important equation in optical physics, crucial for solving problems related to focal length, object distance, and image distance. The formula is given by:\[\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\]where:

• \(f\) is the focal length of the lens.
• \(v\) is the image distance from the lens (the distance where the clear image is formed).
• \(u\) is the object distance from the lens (the distance from the lens to the object being viewed).

This formula allows us to calculate any one of these three variables if the other two are known.
In the original exercise, it helped us determine the new focal length that the man would require as his reading distance shifted. By applying the values given, we used this formula to adjust the focal length appropriately, allowing him to read comfortably again.

Near Point

The near point is the closest distance at which the eye can focus an object clearly. This distance changes with age as the flexibility of the eye's lens decreases, a common issue known as presbyopia.

• For young individuals, the near point is typically around 25 cm.
• As one ages, this point usually moves farther away.

In the exercise's context, the man's near point has moved from 25 cm to 29 cm over five years. This change indicates his need for corrective lenses to maintain the ability to focus up close effectively.
Understanding the concept of the near point is essential for diagnosing vision issues and prescribing appropriate corrective measures, such as the correct focal length of reading glasses or contact lenses.