(a) A gripper is shown in the figure. A horizontal force F = 50 N is applied to the handle of the lever at E. The mean diameter…

(a)

Interpretation. Jaws GAC and BD are pinned to link AB at A and B. The screw is threaded (opposite-hand) into both jaws at C and D (the printed "C and E" is read as C and D, since E is the handle), so turning it drives C and D apart and closes the jaws at G. F acts at right angles to the 125 mm handle.

Torque applied to the screw: M = F × 0.125 = 50 × 0.125 = 6.25 N·m

Screw geometry: Lead angle: tan α = L/(π dₘ) = 5/(π × 25) = 0.06366, so α = 3.643° Friction angle: φs = tan⁻¹ 0.3 = 16.699° Tightening is a "raising-load" case, with tan(φs + α) = tan 20.342° = 0.37074.

Torque equilibrium of the screw. The same axial force W acts at C and at D, and both threads resist rotation: M = 2W(dₘ/2)tan(φs + α) 6.25 = 2 × W × 0.0125 × 0.37074, so W = 674.3 N

Since φs exceeds α, the screw is self-locking, and the clamp holds after the handle is released.

Moment equilibrium of jaw GAC about A. The screw pushes C up with W, the work reacts at G with F_G, and the two-force link AB acts vertically at A.

ΣM_A = 0: F_G × 200 = W × 200, so F_G = W = 674 N

The link AB then carries F_G + W = 1349 N in tension.

The clamping force at G is about 674 N. The equal 200 mm arms pass the screw force to G unchanged. If only one thread resisted the torque, the result would be 1.35 kN.

(b)

Given: P = 125 kN. For aluminium, E = 70 GPa; AB is 300 mm long and 38 mm in diameter, and BC is 450 mm long and 56 mm in diameter. For steel, E = 200 GPa, L = 750 mm and σ_allow = 165 MPa.

Deformation of the aluminium rod: A_AB = π(38)²/4 = 1134.1 mm² and A_BC = π(56)²/4 = 2463.0 mm² δ_Al = (P/E)(L_AB/A_AB + L_BC/A_BC) = (125 000/70 000)(300/1134.1 + 450/2463.0) = 1.7857 × (0.26452 + 0.18270) = 0.799 mm

Criterion 1 (stiffness): δ_st = PL/(A E_st) ≤ 0.799 mm A ≥ 125 000 × 750/(200 000 × 0.7986) = 586.9 mm², so d ≥ √(4 × 586.9/π) = 27.34 mm

Criterion 2 (strength): σ = P/A ≤ 165 MPa A ≥ 125 000/165 = 757.6 mm², so d ≥ √(4 × 757.6/π) = 31.06 mm

Selection. Both conditions must hold, so the larger value governs:

d_min ≈ 31.1 mm, set by the allowable stress.

At this diameter the steel shortens by only 0.62 mm, so strength, not stiffness, controls.

(c)

Given: p = 8 MPa, T = 12 kN·m, dᵢ = 180 mm, t = 12 mm. Since t/dᵢ = 1/15, thin-wall theory applies. The mean radius r = 90 + 6 = 96 mm is used, and the radial stress (zero at the outer surface) is neglected.

Membrane stresses: Hoop: σ_h = pr/t = 8 × 96/12 = 64 MPa Longitudinal (closed ends): σ_l = pr/2t = 32 MPa

Torsional shear (thin tube): τ = T/(2πr²t) = 12 × 10⁶/(2π × 96² × 12) = 17.27 MPa

Principal stresses (plane stress in the wall): σ₁,₂ = (σ_h + σ_l)/2 ± √[((σ_h – σ_l)/2)² + τ²] = 48 ± √(16² + 17.27²) = 48 ± 23.54

σ₁ = 71.5 MPa and σ₂ = 24.5 MPa. The principal planes lie at ½tan⁻¹(2 × 17.27/32) = 23.6° to the hoop direction.

Maximum shear stress: In-plane: τ = (σ₁ – σ₂)/2 = 23.5 MPa With the third principal stress σ₃ ≈ 0 at the outer surface, the absolute maximum is τ_max = σ₁/2 = 35.8 MPa. It acts on planes at 45° through the wall thickness.

Result: maximum normal stress ≈ 71.5 MPa; maximum shearing stress ≈ 35.8 MPa (in-plane 23.5 MPa). Using the inner radius gives 69.7 MPa and 34.9 MPa instead. The torque mainly rotates the principal axes away from the hoop and axial directions.

(d)

The phases of plain-carbon steel, as seen under an optical microscope after nital etching, are sketched below.

(i) Ferrite (α-iron) is an interstitial solid solution of carbon in BCC iron. It dissolves at most about 0.022% C, at 727 °C. In the microscope it shows as light, equiaxed polygonal grains with thin boundaries. It is soft, ductile and ferromagnetic, and it is the main constituent of low-carbon steel.

(ii) Austenite (γ-iron) is an interstitial solid solution of carbon in FCC iron. It dissolves up to about 2.1% C at 1147 °C. In plain-carbon steel it is stable only above 727 °C; at room temperature it survives only in austenitic alloy steels such as 18-8 stainless. It appears as polygonal grains with straight, parallel-sided annealing twins. It is non-magnetic, tough and ductile.

(iii) Cementite (Fe₃C) is an orthorhombic intermetallic compound containing 6.67% C. In hypereutectoid steels (above about 0.8% C) it separates as a white network along the prior austenite grain boundaries, and it also forms the plates within pearlite. It is very hard and brittle.

(iv) Pearlite is the eutectoid mixture formed when austenite of about 0.8% C cools through 727 °C. It consists of alternate lamellae of ferrite (about 88%) and cementite (about 12%) and has a characteristic "mother-of-pearl" look. Its strength and hardness lie between those of ferrite and cementite.

Significance. A hypoeutectoid steel shows ferrite plus pearlite, a eutectoid steel is fully pearlitic, and a hypereutectoid steel shows pearlite with a cementite network. This progression explains why hardness rises and ductility falls as carbon content increases.

(e)

Given: m = 10 mm, φ = 20°, pinion t = 24, speed ratio 2, so gear T = 48. Addendum = 1 module = 10 mm.

Radii: Pinion pitch radius r = mt/2 = 120 mm; gear pitch radius R = mT/2 = 240 mm Addendum radii: r_a = 130 mm; R_a = 250 mm Base radii: r_b = 120 cos 20° = 112.76 mm; R_b = 240 cos 20° = 225.53 mm

(i) Length of path of contact Path of approach (gear addendum) = √(R_a² – R_b²) – R sin φ = 107.879 – 82.085 = 25.794 mm Path of recess (pinion addendum) = √(r_a² – r_b²) – r sin φ = 64.688 – 41.042 = 23.645 mm Path of contact = 25.794 + 23.645 = 49.44 mm

(ii) Contact ratio Arc of contact = path/cos φ = 49.44/0.9397 = 52.61 mm Circular pitch p = πm = 31.416 mm Contact ratio = 52.61/31.416 = 1.67. Equivalently, it is the path divided by the base pitch: 49.44/29.52.

So two pairs of teeth are in contact for about 67% of the time.

(iii) Angle of action of the pinion θ = arc of contact/r = 52.61/120 = 0.4384 rad = 25.12°. Of this, 13.11° is approach and 12.01° is recess.

The contact ratio is comfortably above the usual minimum of about 1.2, so the drive transmits motion smoothly and continuously.