Question W: Derive the dimension of the volume of a cylinder using height (h), radius (r), and diameter (d).

Here are the solutions to the questions: --- Question W: Derive the dimension of the volume of a cylinder using height (h), radius (r), and diameter (d). Step 1: Recall the formula for the volume of a cylinder. The volume of a cylinder (V) is given by: V = r^2 h Step 2: Identify the dimensions of each physical quantity. Radius (r) is a length, so its dimension is [L]. Height (h) is a length, so its dimension is [L]. Diameter (d) is also a length, so its dimension is [L]. is a dimensionless constant. Step 3: Substitute the dimensions into the volume formula. The dimension of V is: [V] = [r]^2 [h] [V] = [L]^2 [L] [V] = [L^3] Step 4: Express the dimension using diameter (d). Since the radius r is half the diameter d (r = (d)/(2)), its dimension is still [L]. Therefore, using diameter instead of radius does not change the overall dimension of volume. [V] = [d]^2 [h] [V] = [L]^2 [L] [V] = [L^3] The dimension of the volume of a cylinder is [L^3]. --- Question 4: If a bob is attached to a string of length 80 cm and lifted at an angle of 45 degrees to cause a displacement. i) Find the speed of the moving bob. Assumption: We assume the question asks for the speed of the bob at the lowest point of its swing after being released from rest at an angle of 45^. We will use the principle of conservation of mechanical energy. Step 1: Convert the string length to meters. Length of the string, L = 80 \, cm = 0.80 \, m. Step 2: Calculate the vertical height (h) the bob falls. When the bob is lifted to an angle , its vertical height from the pivot is L . When it is at the lowest point, its vertical height from the pivot is L. The change in vertical height (the height it falls) is: h = L - L = L(1 - ) Given = 45^: h = 0.80 \, m (1 - 45^) h = 0.80 \, m (1 - sqrt(2)2) h = 0.80 \, m (1 - 0.7071) h = 0.80 \, m (0.2929) h = 0.23432 \, m Step 3: Apply the conservation of mechanical energy. At the initial position (lifted at 45^), the bob has potential energy (PE_i) and zero kinetic energy (KE_i = 0). At the lowest point, the bob has zero potential energy (PE_f = 0) and maximum kinetic energy (KE_f). PE_i + KE_i = PE_f + KE_f mgh + 0 = 0 + (1)/(2)mv^2 Where m is the mass of the bob, g is the acceleration due to gravity (9.8 \, m/s^2), and v is the speed at the lowest point. Step 4: Solve for the speed (v). mgh = (1)/(2)mv^2 Divide both sides by m: gh = (1)/(2)v^2 v^2 = 2gh v = sqrt(2gh) Substitute the values for g and h: v = sqrt(2 × 9.8 \, m/s)^2 × 0.23432 \, m v = sqrt(4.592672 \, m)^2/s^2 v ≈ 2.143 \, m/s The speed of the moving bob at the lowest point is 2.14 \, m/s. ii) Calculate X from the diagram shown. The diagram mentioned in this part of the question is not provided. Therefore, it is not possible to calculate X.