Determine the dimensions of the Universal Gravitational Constant G and Pressure.

Question 7: Dimensions of Universal Gravitational Constant G Step 1: Recall Newton's Law of Universal Gravitation. The formula for the gravitational force (F) between two masses (m_1, m_2) separated by a distance (r) is: F = G (m_1 m_2)/(r^2) Where G is the universal gravitational constant. Step 2: Rearrange the formula to solve for G. G = (F r^2)/(m_1 m_2) Step 3: Determine the dimensions of each term. • Dimensions of Force (F): M L T^-2 • Dimensions of Distance (r): L • Dimensions of Mass (m_1, m_2): M Step 4: Substitute the dimensions into the equation for G. [G] = [F] [r]^2[m_1] [m_2] [G] = (M L T^-2) (L)^2(M) (M) [G] = M L^3 T^-2M^2 Step 5: Simplify the dimensions. [G] = M^1-2 L^3 T^-2 [G] = M^-1 L^3 T^-2 The correct combination of dimensions for the Universal Gravitational Constant G is a) M^-1L^3T^-2. --- Question 8: Dimensions of Pressure Step 1: Recall the definition of Pressure. Pressure (P) is defined as force (F) per unit area (A): P = (F)/(A) Step 2: Determine the dimensions of Force and Area. • Dimensions of Force (F): M L T^-2 • Dimensions of Area (A): L^2 Step 3: Substitute the dimensions into the equation for Pressure. [P] = [F][A] [P] = M L T^-2L^2 Step 4: Simplify the dimensions. [P] = M L^1-2 T^-2 [P] = M L^-1 T^-2 The correct combination of dimensions for Pressure is a) ML^-1T^-2. --- Question 9: Original length of the pendulum Step 1: Recall the formula for the period of a simple pendulum. The period (T) of a simple pendulum is given by: T = 2 sqrt((L)/(g)) Where L is the length of the pendulum and g is the acceleration due to gravity. Step 2: Set up equations for the initial and final states. Let T_1 be the initial period and L_1 be the original length. Let T_2 be the new period and L_2 be the new length. Given: • T_1 = 7.0 s • The length is shortened by 1.5 m, so L_2 = L_1 - 1.5 m. • The new period is T_2 = 8.5 s. The equations are: T_1 = 2 sqrt((L_1)/(g)) (1) T_2 = 2 sqrt((L_2)/(g)) (2) Step 3: Form a ratio of the periods to eliminate constants. Divide equation (1) by equation (2): (T_1)/(T_2) = (2 sqrt(L_1)/(g))2 sqrt((L_2)/(g)) (T_1)/(T_2) = sqrt((L_1)/(L_2)) Step 4: Substitute the given values and solve for L_1. Square both sides: ((T_1)/(T_2))^2 = (L_1)/(L_2) Substitute T_1 = 7.0 s, T_2 = 8.5 s, and L_2 = L_1 - 1.5 m: (7.0 s8.5 s)^2 = (L_1)/(L_1 - 1.5 m) ((7)/(8.5))^2 = (L_1)/(L_1 - 1.5) (49)/(72.25) = (L_1)/(L_1 - 1.5) Cross-multiply: 49(L_1 - 1.5) = 72.25 L_1 49 L_1 - 49 × 1.5 = 72.25 L_1 49 L_1 - 73.5 = 72.25 L_1 Rearrange to solve for L_1: -73.5 = 72.25 L_1 - 49 L_1 -73.5 = 23.25 L_1 L_1 = (-73.5)/(23.25) L_1 ≈ -3.16 m *Note: The calculated original length is negative, which is physically impossible for a pendulum. This indicates an inconsistency in the problem statement, as shortening the length of a pendulum should decrease its period, but here the period increased