MECHANICS OF MATERIALS
COMPREHENSIVE PAPER 2026 QUESTIONS
AND SOLUTIONS GRADED A+
◉Polar moment of inertia for a circle of radius c. Answer:
J=1/2(πc^4)
◉Formula for angle of twist Φ. Answer: Φ=TL/JG
T is the torque
L is the length
J is the polar moment of inertia
G is the modulus of shearing elasticity
◉Total angle of twist in a rod of variable circular cross section (and
thus radius). Answer: Φ=S_0^L Tdx/JG
◉Equation for torque of a rigid beam. Answer: T=P/(2*pi*f)
◉Maximum allowable radius for the shaft based on the maximum
shearing stress. Answer: c=(J/T)*τ_max
J is the polar moment of inertia
, ◉What is poisson's ratio?. Answer: v=-(lateral strain )/(axial strain);
theoretical range of 0.0 to 0.5
Lateral strain is how much the material is stretched perpendicular to
the applied load, axial strain is how much the material is stretched in
the axial direction when the load is applied.
◉Polar moment of inertia for a hollow shaft. Answer:
J/c_2=T/τ_max=pi/((2c_2)(c_2^4-c_1^4))
◉Formula for gear ratios. Answer: Gear ratio=D_1/D_2 (the larger
diameter divided by the smaller diameter). This gives the ratio of
rotations between gears.
◉Definition of pure bending. Answer: Prismatic members subject to
equal and opposite couples acting in the same longitudinal plane.
◉Sum of forces along ends of beams in pure bending. Answer:
F_x=Sσ_xdA=0
The sum of all normal stresses along a bar in pure bending is zero
because those above and below the neutral axis net to zero. Those
above the NA are compressive, those below are tensile, for a beam
bending CC up.
◉Sum of moments of a bar in pure bending (y axis). Answer: ΣMy=S
z*σ_x dA=0
COMPREHENSIVE PAPER 2026 QUESTIONS
AND SOLUTIONS GRADED A+
◉Polar moment of inertia for a circle of radius c. Answer:
J=1/2(πc^4)
◉Formula for angle of twist Φ. Answer: Φ=TL/JG
T is the torque
L is the length
J is the polar moment of inertia
G is the modulus of shearing elasticity
◉Total angle of twist in a rod of variable circular cross section (and
thus radius). Answer: Φ=S_0^L Tdx/JG
◉Equation for torque of a rigid beam. Answer: T=P/(2*pi*f)
◉Maximum allowable radius for the shaft based on the maximum
shearing stress. Answer: c=(J/T)*τ_max
J is the polar moment of inertia
, ◉What is poisson's ratio?. Answer: v=-(lateral strain )/(axial strain);
theoretical range of 0.0 to 0.5
Lateral strain is how much the material is stretched perpendicular to
the applied load, axial strain is how much the material is stretched in
the axial direction when the load is applied.
◉Polar moment of inertia for a hollow shaft. Answer:
J/c_2=T/τ_max=pi/((2c_2)(c_2^4-c_1^4))
◉Formula for gear ratios. Answer: Gear ratio=D_1/D_2 (the larger
diameter divided by the smaller diameter). This gives the ratio of
rotations between gears.
◉Definition of pure bending. Answer: Prismatic members subject to
equal and opposite couples acting in the same longitudinal plane.
◉Sum of forces along ends of beams in pure bending. Answer:
F_x=Sσ_xdA=0
The sum of all normal stresses along a bar in pure bending is zero
because those above and below the neutral axis net to zero. Those
above the NA are compressive, those below are tensile, for a beam
bending CC up.
◉Sum of moments of a bar in pure bending (y axis). Answer: ΣMy=S
z*σ_x dA=0
