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.
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You're on a roll — Question 6: The question asks what can cause a change in a vector. Step 1: Consider the effect of "Rotation of frame of reference". If the coordinate system (frame of reference) rotates, the components of a vector, when expressed in that frame, will change. For example, a vector pointing along the X-axis in one frame will have both X and Y components in a frame rotated by 45^. Although the vector itself (its magnitude and direction in absolute space) remains unchanged, its representation (components) in the rotated frame changes. In physics, a change in components is considered a "change in a vector" in the context of its representation. Step 2: Consider the effect of "Rotation of vector". If the vector itself rotates, its direction changes. A vector is defined by both its magnitude and direction. If either of these changes, the vector itself changes. For example, if a vector pointing east rotates to point north, it is a different vector. Step 3: Conclude based on both effects. Both the rotation of the frame of reference (leading to a change in components) and the rotation of the vector itself (leading to a change in direction) result in a change in the vector. The final answer is C. Question 7: Given that the resultant of two vectors A and B is perpendicular to A and its magnitude is half of B. We need to find the angle between A and B. Step 1: Define the resultant vector and use the given conditions. Let R = A + B be the resultant vector. Given that R is perpendicular to A, their dot product is zero: R · A = 0 (A + B) · A = 0 A · A + B · A = 0 Let A = |A| and B = |B|. Let be the angle between A and B. A^2 + AB = 0 Since A 0 (otherwise R would be B and not perpendicular to A unless B is also zero), we can divide by A: A + B = 0 A = -B (1) Since A is a magnitude, A > 0. This implies that must be negative, so is an obtuse angle. Step 2: Use the given magnitude of the resultant. Given that the magnitude of the resultant is half of B: |R| = (1)/(2)|B| R = (B)/(2) The magnitude of the resultant of two vectors is given by: R^2 = A^2 + B^2 + 2AB Substitute R = B/2: ((B)/(2))^2 = A^2 + B^2 + 2AB (B^2)/(4) = A^2 + B^2 + 2AB (2) Step 3: Substitute equation (1) into equation (2) to solve for . From (1), A = -B. Substitute this into (2): (B^2)/(4) = (-B)^2 + B^2 + 2(-B)B (B^2)/(4) = B^2^2 + B^2 - 2B^2^2 (B^2)/(4) = B^2 - B^2^2 Assuming B 0, divide the entire equation by B^2: (1)/(4) = 1 - ^2 Rearrange to solve for ^2: ^2 = 1 - (1)/(4) ^2 = (3)/(4) Take the square root: = ±sqrt(3)2 From Step 1, we determined that must be negative. = -sqrt(3)2 The angle in the range [0, ] for which = -sqrt(3)2 is: = 150^ or (5)/(6) radians The final answer is D. Question 8: Given A = B + C and the magnitudes A=5, B=4, C=3. We need to find the angle between A and C. Step 1: Determine the relationship between B and C. We are given the magnitudes A=5, B=4, C=3. Notice that 3^2 + 4^2 = 9 + 16 = 25 = 5^2. This means C^2 + B^2 = A^2. For vector addition A = B + C, the magnitude is given by A^2 = B^2 + C^2 + 2BC, where is the angle between B and C. Substituting the magnitudes: 5^2 = 4^2 + 3^2 + 2(4)(3) 25 = 16 + 9 + 24 25 = 25 + 24 0 = 24 = 0 This implies = 90^ or ()/(2) radians. Therefore, vector B is perpendicular to vector C. Step 2: Find the angle between A and C. Let be the angle between A and C. We can use the dot product definition: A · C = |A||C| Substitute A = B + C: (B + C) · C = AC B · C + C · C = AC Since B is perpendicular to C, B · C = 0. Also, C · C = |C|^2 = C^2. 0 + C^2 = AC C^2 = AC Since C 0, we can divide by C: C = A = (C)/(A) Substitute the given magnitudes C=3 and A=5: = (3)/(5) = ^-1((3)/(5)) The final answer is B. Drop the next question.