This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Define the mirror formula and magnification. For a concave mirror, the mirror formula is given by: (1)/(f) = (1)/(u) + (1)/(v) where f is the focal length, u is the object distance, and v is the image distance. For real images, u and v are positive. The linear magnification M is given by: M = image heightobject height = (v)/(u) Since the image is real, it is inverted, so the magnification is typically negative. However, when referring to the "size" of the image, we use the absolute value of magnification. Step 2: Analyze the initial state. In the initial state, the image is 3 times the size of the object. So, M_1 = (v_1)/(u_1) = 3. This implies v_1 = 3u_1. Substitute this into the mirror formula: (1)/(f) = (1)/(u_1) + (1)/(3u_1) (1)/(f) = (3)/(3u_1) + (1)/(3u_1) = (3+1)/(3u_1) = (4)/(3u_1) f = (3u_1)/(4) (1) Step 3: Analyze the final state. In the final state, the image is 5 times the size of the object. So, M_2 = (v_2)/(u_2) = 5. This implies v_2 = 5u_2. Substitute this into the mirror formula: (1)/(f) = (1)/(u_2) + (1)/(5u_2) (1)/(f) = (5)/(5u_2) + (1)/(5u_2) = (5+1)/(5u_2) = (6)/(5u_2) f = (5u_2)/(6) (2) Step 4: Relate the initial and final states using the screen shift. Equate the expressions for f from (1) and (2): (3u_1)/(4) = (5u_2)/(6) Multiply both sides by 12 to clear denominators: 3(3u_1) = 2(5u_2) 9u_1 = 10u_2 u_2 = (9)/(10)u_1 (3) Since the magnification increased from 3 to 5, the object must have moved closer to the mirror, and the image moved further away. Therefore, u_2 < u_1 and v_2 > v_1. The shift of the screen (image) is given as 30 cm: v_2 - v_1 = 30 cm Substitute v_1 = 3u_1 and v_2 = 5u_2: 5u_2 - 3u_1 = 30 (4) Step 5: Solve for u_1 and u_2. Substitute equation (3) into equation (4): 5((9)/(10)u_1) - 3u_1 = 30 (45)/(10)u_1 - 3u_1 = 30 (9)/(2)u_1 - 3u_1 = 30 (9u_1 - 6u_1)/(2) = 30 (3u_1)/(2) = 30 3u_1 = 60 u_1 = 20 cm Now find u_2 using equation (3): u_2 = (9)/(10)(20 cm) = 18 cm Step 6: Calculate the focal length and the shift of the object. Calculate the focal length f using u_1 in equation (1): f = (3u_1)/(4) = 3(20 cm)4 = 60 cm4 = 15 cm The shift of the object is u = u_1 - u_2: u = 20 cm - 18 cm = 2 cm The focal length of the mirror is 15 cm. The shift of the object is 2 cm.