(a) An automobile weighing 15 kN is moving at a speed of 100 km per hour. The dynamic coefficient of friction between the rubber…

(a) Given W = 15 kN, v = 100 km/h = 250/9 m/s = 27.778 m/s, μ = 0.60. Use work-energy: loss of kinetic energy equals work done against resistance. Since W = mg, weight cancels and distance is independent of the vehicle weight.

(a)(i) Flat road: Normal reaction N = W, friction = μW, so deceleration a = μg. s = v²/(2μg) = (250/9)²/(2 × 0.60 × 9.81) = 65.55 m. Distance = 65.55 m.

(a)(ii) Road inclined 7° upward: N = W cos θ. Retarding force = μW cos θ + W sin θ. a = g(μ cos θ + sin θ) = 9.81(0.60 cos 7° + sin 7°) = 7.038 m/s². s = v²/(2a) = (250/9)²/(2 × 7.038) = 54.82 m. Distance = 54.82 m.

(a)(iii) Road inclined 7° downward: Gravity assists motion, friction opposes it. a = g(μ cos θ − sin θ) = 9.81(0.60 cos 7° − sin 7°) = 4.647 m/s². s = v²/(2a) = (250/9)²/(2 × 4.647) = 83.03 m. Distance = 83.03 m. Validity: μ cos θ > sin θ; here 0.5955 > 0.1219, so the car stops.

(b) Annealing is a heat-treatment process in which steel is heated to or near the austenitizing temperature, held long enough for homogenisation, and then cooled very slowly, usually in the furnace. Its principle is best seen on a CCT diagram.

A CCT diagram plots temperature against time on a logarithmic scale and shows the start and finish curves for diffusion-controlled transformations such as ferrite, pearlite and bainite, and the martensite start and finish temperatures. For annealing, the cooling curve is made to lie far to the right of the pearlite start and finish curves. Therefore austenite transforms at a relatively high temperature by diffusion into ferrite and pearlite, and the cooling curve avoids the martensite region altogether. The resulting structure is near-equilibrium, coarse, soft and ductile. If subcritical spheroidizing annealing is used, cementite is eventually obtained as spheroids in a ferrite matrix.

The purpose of annealing is:

• to soften the metal and improve machinability;
• to relieve internal stresses produced by casting, forging, welding or cold working;
• to improve ductility and toughness;
• to homogenise the microstructure and remove segregation;
• to refine grain size in some full-annealing operations;
• to prepare the material for subsequent cold working or machining.

Applications include annealing of low- and medium-carbon steel castings and forgings, welded joints, cold-worked sheets and wires, and spheroidizing high-carbon or alloy steels before machining. Stress-relief annealing is widely used after welding, and recrystallization annealing is used after cold deformation to restore ductility.

(c) Let plane B be the datum, θ_B = 0°, θ_C = 90°, θ_D = 210° in the counterclockwise direction. Let F = mr in kg·mm.

F_B = 15 × 480 = 7200 kg·mm at 0°. F_C = 25 × 240 = 6000 kg·mm at 90°. F_D = 20 × 300 = 6000 kg·mm at 210°.

For complete balance, primary force balance gives ΣF = 0. F_B + F_C + F_D = (7200 − 3000√3) + i(6000 − 3000) = (7200 − 3000√3) + i3000. Therefore F_A = −[(7200 − 3000√3) + i3000] = (3000√3 − 7200) − i3000.

(c)(i) Magnitude: F_A = √[(7200 − 3000√3)² + 3000²] = 3607.69 kg·mm. m_A = F_A/r_A = 3607.69/360 = 10.02 kg.

Angular position: the vector F_A has both components negative, so it lies in the third quadrant. θ_A = 180° + tan⁻¹[3000/(7200 − 3000√3)] = 236.26° from B in the counterclockwise direction. Mass A = 10.02 kg, angular position = 236.26° CCW from B.

(c)(ii) Take moments about plane B. Let z be positive from B toward C, so z_B = 0, z_C = 300 mm. Couple balance: F_A z_A + F_C(300) + F_D z_D = 0.

Using components: F_A = (−2003.85, −3000), F_C = (0, 6000), F_D = (−5196.15, −3000).

From x-components: 2003.85 z_A + 5196.15 z_D = 0. From y-components: −3000 z_A − 3000 z_D + 6000 × 300 = 0, so z_A + z_D = 600 mm.

Solving: z_D = −376.63 mm, z_A = 976.63 mm.

Taking B as datum with C at +300 mm, plane A is at z_A = +976.63 mm and plane D is at z_D = −376.63 mm. Thus A is 976.63 mm from B on the C-side, i.e. 676.63 mm beyond C; D is 376.63 mm from B on the side opposite to C, i.e. 676.63 mm from C on the other side from A.