Direct Multiplication Method - General Physics - Solved Past Paper, Exams of Physics

Download Direct Multiplication Method - General Physics - Solved Past Paper and more Exams Physics in PDF only on Docsity! 3. (10 pts) Compute the torque on an object if a force of ~F = −3 ı̂ + 7 ̂ − 4k̂ N is exerted at the position ~r = 6 ı̂ − 10̂ + 9k̂ m. You can either multiply out all 9 terms and then work the cross-products of the unit vectors or use the determinant method. Both techniques give the same answer. First, the direct multiplication method: ~τ = ( 6 ı̂ − 10 ̂ + 9 k̂ ) × ( −3 ı̂ + 7 ̂ − 4 k̂ ) = −18 (̂ı × ı̂)+42 (̂ı × ̂)−24 ( ı̂ × k̂ ) +30 (̂ × ı̂)−70 (̂ × ̂)+40 ( ̂ × k̂ ) −27 ( k̂ × ı̂ ) +63 ( k̂ × ̂ ) −36 ( k̂ × k̂ ) = 42 k̂ + 24 ̂− 30 k̂ + 40 ı̂− 27 ̂− 63 ı̂ ~τ = −23 ı̂− 3 ̂ + 12 k̂ N · m And then the determinant method: ~τ = ∣ ∣ ∣ ∣ ∣ ∣ ı̂ ̂ k̂ 6 −10 9 −3 7 −4 ∣ ∣ ∣ ∣ ∣ ∣ = ı̂(40 − 63) − ̂(−24 + 27) + k̂(42 − 30) ~τ = −23 ı̂− 3 ̂ + 12 k̂ N · m There was a lot of very sloppy notation used in this problem. The most prevalent was a lack of vector signs, like: τ = r × F which is not correct! The magnitude of the torque is not equal to the product of the magnitudes of r and F . You must also multiply by the sin of the angle between them! However, with the vector symbols it is correct, ~τ = ~r × ~F Also, there were a lot of vector components written with i, j, and k rather than ı̂, ̂, or k̂. i, j, and k are just (scalar) variables unless you specifically indicate, with the hat symbol, that they are unit vectors. Another notational mistake occurred in the actual vector multiplication. Torque is a vector, so the type of product must have a vector as a result. Therefore it cannot be the result of a dot product! So ~τ = ~r · ~F is completely wrong. You could compute the dot product of ~r and ~F , but it does not give anything related to the torque. I also saw the expression written as ~τ = ~r ~F or even with the components of the two specified vectors, ~τ = ( 6 ı̂ − 10 ̂ + 9 k̂ ) ( −3 ı̂ + 7 ̂ − 4 k̂ ) (with or without appropriate hats on the unit vectors) but with no multiplication symbol. For multiply- ing vectors, you must specify which multiplication you intend to use, otherwise it does not make sense. ~A ~B is not defined. You can’t do it. While I did not subtract any points for these notational errors, I did take off points if the order of the vectors in the cross product was reversed, ~F × ~r, since this gives −~τ rather than ~τ .