Solve Prob. 1.15 graphically.

Name: Problem 16, Chapter 1: Fundamentals of Optics
Uploaded: 2022-01-26T14:36:32-08:00
Duration: 10 min 50 s
Channel: Mohammed Nadhir
Description: Solve Prob. 1.15 graphically.

Key Concepts

Coordinate Plane

The coordinate plane, defined by its x-axis and y-axis, is essential for graphically representing functions. It provides a structured framework for plotting points, curves, and other geometric figures, thereby enabling accurate visualization and analysis of mathematical relationships and solutions.

Graphical Method

The graphical method involves solving equations by plotting their corresponding functions on a coordinate plane and visually identifying the solution as the point or points where the graphs intersect. This approach allows one to gain intuitive insight into the behavior of functions and their relationships, providing a visual representation of the solution that complements analytical techniques.

Intersection Points

Intersection points refer to the coordinates on the graph where two or more function curves cross each other. In the context of solving equations graphically, these points represent the values that satisfy all of the equations simultaneously, as each intersection corresponds to a solution of the system.

Transcript

00:01 So here the question asked is to solve the problem 6 .5 graphically and the problem 6 .5 was the radius of a spherical mirror is plus 18 cm.

00:11 Now from that plus value we can understand that it is a convex mirror because for convex mirrors the radius of curvature will be equal to will be positive values.

00:23 So the radius of curvature is given as plus 18 centimeter.

00:27 Now, it is told that an object 4 centimeter high is located in front of the mirror.

00:34 So, the height of the object is, that is, i'm writing in edge, that is 4 centimeters, and it is given that the object is placed in front of the mirror at various distances, that is the a value is 36 centimeter.

00:49 So question a is 36 centimeters.

00:52 Question b is 24 and c is 12.

00:57 B is 24 centimeters and c is 12 centimeters.

01:03 Now if we draw the figure of a convex mirror, it will be like this because for convex mirrors, the smooth surface will be towards this side of the mirror and the rough surface will be towards this side of the mirror.

01:19 Now, since the radius of curvature is positive, we'll be drawing or we know that the radius of curvature and focus all will be to the right side of the mirror.

01:29 So we can mark this value.

01:31 This will be the point p or the pole and from pole to center of curvature the value was 18 centimeter.

01:40 It is given and we know that the value of focal distance that is small f will be equal to r by two which will be equal to nine centimeter so from pole to focus from ball to focus this distance is nine centimeter now we would have to find where the image is forming and the height of the image now there are two formulas for that that is the mirror formula that is one by v plus one by u equal to one by f here v means image image distance and u means object distance and f is the focal length and this is the one important formula that we will be using to solve all the parts of the question and the second part or the second formula is height of the object divided by height of image will be equal to the object distance divided by the image distance.

02:53 So here it is xo, that is height of object in numerator.

02:57 So again, to the other side of the numerator will be also the object distance.

03:01 So from these two formulas, we can just find or we can just answer the question asked over here.

03:09 Now, the first question or the a part was when the object distance is 36 centimeters, that is u equal to 36 centimeter now one thing that we would have to be very careful here is the object is placed in front of the mirror which means to the left side of the mirror so that this value will be negative i'm just marking this with this with a different color so the images to the left side will be taking as negative and sorry the distances to the left side will be taking as negative and the distances to the right side will be taking as positive that's why we have taken this value of radius of curvature as positive.

03:49 Now, since the object is placed to the left side of the mirror, which means the value of you must be taken as negative.

03:59 Now, from the formula that we have, we can write 1 by v plus 1 by minus 36 equal to 1 by the value of f is 9, which means 1 by v equal to 1 by 9 plus 1 by 36 which means value of v will be equal to 36 into 9 divided by 36 plus 9 which means the value of we can be found as 7 .2 centimeters 7 .2 centimeters so from this figure we can just mark this value from here to here this is the object or sorry this is the image that is formed so this value is 7 .2 centimeters.

04:53 Now one thing we can understand or there is one law that is for convex mirrors and if the distance is, sorry, the object is placed in front of the mirror in a specific distance or a distance that we know that we know then the image formed will be between the pole and the focus.

05:15 Just like this over here.

05:16 So we know that this object, sorry, this image is formed in between the focus and the board.

05:21 So that will be like that always.

05:23 Now, the next thing that we would have to find is the height of the image.

05:29 So to find the height of image, we have the value of height of object, which was equal to four centimeter.

05:34 So four divided by height of image we don't know, so that is minus hi equal to the value of you.

05:41 We know that was minus 36.

05:44 We would have to give this as minus 36 divided by the value of v was 7 .2 which means the height of the object image is equal to 7 .2 into 4 divided by minus 36 which is equal to sorry for a small mistake over here this equation is having an already a negative over here so this equation will be having one more negative over here this will be having one more negative over here which will be like this so this value equal to 7 .2 into 4...

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