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
Here are the calculations for each part: a. For r = (4,5,3) and p = (1,4,-2) Step 1: Calculate the angular momentum L. The angular momentum L is given by the cross product of the position vector r and the linear momentum vector p: L = r × p = i & j & k \\ 4 & 5 & 3 \\ 1 & 4 & -2 L = (5(-2) - 3(4))i - (4(-2) - 3(1))j + (4(4) - 5(1))k L = (-10 - 12)i - (-8 - 3)j + (16 - 5)k L = -22i + 11j + 11k The angular momentum is L = (-22, 11, 11). Step 2: Find the momentum vector and the angle between r and p. The momentum vector is given as p = (1,4,-2). To find the angle between r and p, we use the dot product formula: = r · p|r| |p| First, calculate the dot product: r · p = (4)(1) + (5)(4) + (3)(-2) = 4 + 20 - 6 = 18 Next, calculate the magnitudes of r and p: |r| = sqrt(4^2 + 5^2 + 3^2) = sqrt(16 + 25 + 9) = sqrt(50) |p| = sqrt(1^2 + 4^2 + (-2)^2) = sqrt(1 + 16 + 4) = sqrt(21) Now, substitute these values into the cosine formula: = (18)/(sqrt(50) 21) = (18)/(sqrt(1050)) = ((18)/(sqrt(1050))) ≈ (0.55549) ≈ 56.26^ The angle between the position and momentum vectors is ≈ 56.26^. b. For r = (1,-2,3) and p = (7,-1,1) Step 1: Calculate the angular momentum L. L = r × p = i & j & k \\ 1 & -2 & 3 \\ 7 & -1 & 1 L = ((-2)(1) - 3(-1))i - (1(1) - 3(7))j + (1(-1) - (-2)(7))k L = (-2 + 3)i - (1 - 21)j + (-1 + 14)k L = 1i + 20j + 13k The angular momentum is L = (1, 20, 13). Step 2: Find the momentum vector and the angle between r and p. The momentum vector is given as p = (7,-1,1). First, calculate the dot product: r · p = (1)(7) + (-2)(-1) + (3)(1) = 7 + 2 + 3 = 12 Next, calculate the magnitudes of r and p: |r| = sqrt(1^2 + (-2)^2 + 3^2) = sqrt(1 + 4 + 9) = sqrt(14) |p| = sqrt(7^2 + (-1)^2 + 1^2) = sqrt(49 + 1 + 1) = sqrt(51) Now, substitute these values into the cosine formula: = (12)/(sqrt(14) 51) = (12)/(sqrt(714)) = ((12)/(sqrt(714))) ≈ (0.44983) ≈ 63.27^ The angle between the position and momentum vectors is ≈ 63.27^. c. For r = (0,2,0) and p = (1,0,0) Step 1: Calculate the angular momentum L. L = r × p = i & j & k \\ 0 & 2 & 0 \\ 1 & 0 & 0 L = (2(0) - 0(0))i - (0(0) - 0(1))j + (0(0) - 2(1))k L = (0 - 0)i - (0 - 0)j + (0 - 2)k L = 0i + 0j - 2k The angular momentum is L = (0, 0, -2). Step 2: Find the momentum vector and the angle between r and p. The momentum vector is given as p = (1,0,0). First, calculate the dot product: r · p = (0)(1) + (2)(0) + (0)(0) = 0 + 0 + 0 = 0 Since the dot product is 0, the vectors r and p are orthogonal. = (0)/(|r)| |p| = 0 = (0) = 90^ The angle between the position and momentum vectors is = 90^. That's 2 down. 3 left today — send the next one.