Which of the following quantities is dimensionless? A) Velocity gradient B) Pressure gradient C) Displacement gradient D) Force

Here are the solutions to the problems: Question 11: To determine which quantity is dimensionless, we find the dimensions of each option. A) Velocity gradient: Velocity has dimensions [L T^-1]. Gradient implies division by length, so its dimensions are [L T^-1][L] = [T^-1]. This is not dimensionless. B) Pressure gradient: Pressure (Force/Area) has dimensions [M L T^-2][L^2] = [M L^-1 T^-2]. Pressure gradient has dimensions [M L^-1 T^-2][L] = [M L^-2 T^-2]. This is not dimensionless. C) Displacement gradient: Displacement has dimensions [L]. Displacement gradient has dimensions ([L])/([L]) = [M^0 L^0 T^0]. This is dimensionless. D) Force: Force has dimensions [M L T^-2]. This is not dimensionless. Therefore, the only dimensionless quantity among the options is displacement gradient. The final answer is C Question 12: Step 1: Determine the dimensions of Energy (Joule). Energy (Work) = Force × Displacement Force = mass × acceleration = [M] × [L T^-2] = [M L T^-2] Energy = [M L T^-2] × [L] = [M L^2 T^-2] Step 2: Set up the conversion between the old (SI) system and the new system. Let n_1 be the numerical value in the old system and u_1 be the unit in the old system. Let n_2 be the numerical value in the new system and u_2 be the unit in the new system. We have n_1 u_1 = n_2 u_2. Here, n_1 = 1 (for 1 Joule). The unit in the old system is u_1 = (1 kg) (1 m)^2 (1 s)^-2. The unit in the new system is u_2 = ( kg) ( m)^2 ( s)^-2. Step 3: Calculate n_2. n_2 = n_1 (u_1)/(u_2) = 1 × [M_1 L_1^2 T_1^-2][M_2 L_2^2 T_2^-2] n_2 = ((M_1)/(M_2)) ((L_1)/(L_2))^2 ((T_1)/(T_2))^-2 Substitute the given values: M_1 = 1 kg, M_2 = kg L_1 = 1 m, L_2 = m T_1 = 1 s, T_2 = s n_2 = (1 kg kg) (1 m m)^2 (1 s s)^-2 n_2 = ((1)/()) ((1)/())^2 ((1)/())^-2 n_2 = (1)/() × (1)/(^2) × ^2 n_2 = (^2)/( ^2) So, 1 Joule is equivalent to (^2)/( ^2) in the new system of units. The final answer is A