Exercise 1.9

Transcript

Exercise 1.9 The photograph of a house occupies an area of 1.75 cm² on a 35 mm slide. The slide is projected on to a screen, and the area of the house on the screen is 1.55 m². What is the linear magnification of the projector-screen arrangement. First lets see how Linear Magnification differs from Area Magnification Linear magnification vs Area magnification Image Object f Both dimensions x2 lcm <-> lcm 2cm Area = 1cm? -Area = 4cm? LINEAR MAGNIFICATION AREA MAGNIFICATION Compares corresponding lengths Compares areas __ image height M, = image area ~ object height A ~ object area 2 4 i 2x = 4x Twice the length x twice the width = four times the area Area magnification = (linear magnification)? For equal scaling in both dimensions. Given: 𝐴_(slide )=1.75 cm^(2) 𝐴_(screen )=1.55 m^(2) First converting the area into same units 𝑨_(screen )=1.55 m^(2) =1.55 × (1 m)^(2) =1.55 × (100 cm)^(2) =𝟏.𝟓𝟓 × 𝟏𝟎^(𝟒) cm^(2) Now, If both length and breadth increase by 𝑀 (M is linear magnification): 𝐴_(image )=(𝑀𝑙)(𝑀𝑏) 𝐴_(image ) =𝑀^(2)𝑙𝑏 𝐴_(image ) =𝑀^(2)𝑙𝑏 Thus: 𝑀=sqrt((𝐴_(image ))/(𝐴_(object ))) First convert the areas into the same units: 𝐴_(slide )=1.75×10^(−4) m^(2) Then: [𝑀=sqrt((1.55)/(1.75×10^(−4))); 𝑀=sqrt(8857.142857…)=94.1123948…; 𝑀=94.1] Significant figures: both areas have three significant figures, so the magnification is reported to three significant figures. Common mistake: the area ratio, approximately 8857 , is not the linear magnification. Its square root is required. 𝐴_(image ) =𝑀^(2)𝑙𝑏 𝐴_(image ) =𝑀^(2) × 𝐴_(𝑂𝑏𝑗𝑒𝑐𝑡) Thus: Linear Magnification = 𝑀=sqrt((𝐴_(𝐼𝑚𝑎𝑔𝑒))/(𝐴_(𝑂𝑏𝑗𝑒𝑐𝑡))) 𝑴=sqrt((𝟏.𝟓𝟓 × 𝟏𝟎^(𝟒))/(𝟏.𝟕𝟓)) 𝑀=sqrt(8857.142857…) 𝑀=94.1123948… Significant figures: Since both areas have three significant figures, so the magnification will have three significant figures Rounding off 94.1123948… to 94.1 Thus, Linear Magnification = 94.1 Common mistake: The area ratio, approximately 8857 , is not the linear magnification. Its square root is required. Rounding off 94.1123948… to 94.1 Thus, Linear Magnification = 94.1 Common mistake: The area ratio, approximately 8857 , is not the linear magnification. Its square root is required.