(II) Consider the track shown in Fig. 8–39. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction μₖ = 0.25 . The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine:

(a) the velocity of the block at point B ;

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Verified step by step guidance

1. Step 1: Apply the conservation of mechanical energy between points A and B. Since the track is frictionless between these points, the mechanical energy (sum of potential and kinetic energy) at A will equal the mechanical energy at B. Use the equation: \(E_{A} = E_{B}\), where \(E = K + U\), K is kinetic energy and U is potential energy.
2. Step 2: Calculate the potential energy at A and B. At A, the block has maximum potential energy and no kinetic energy since it starts from rest. The potential energy at A is given by \(U_{A} = mgh\), where \(h\) is the height of point A from B. At B, the height is zero, so \(U_{B} = 0\).
3. Step 3: Set up the energy conservation equation from step 1, substituting the expressions for kinetic and potential energy. Solve for the velocity at B using \(v_{B} = \sqrt{2gh}\), where \(g\) is the acceleration due to gravity.
4. Step 4: Analyze the motion from B to C where friction is involved. Use the work-energy principle, which states that the work done by non-conservative forces (like friction) plus the initial mechanical energy equals the final mechanical energy. Calculate the work done by friction using \(W_{\text{friction}} = -\mu_{k} mg d\), where \(d\) is the distance from B to C.
5. Step 5: Apply the work-energy principle to find the velocity at C. Use the equation \(K_{B} + W_{\text{friction}} = K_{C}\) to find the kinetic energy at C, and then solve for the velocity at C using \(v_{C} = \sqrt{\frac{2K_{C}}{m}}\).