Rotation
ω = V/r ω is angular velocity in radians/sec 1 radian/sec = 9.55 rev/min r is radius of circle in meters V is the tangental velocity in m/s Angular position θ = ωt Centripetal force f = mV²/r = mrω² ω is angular velocity in radians/sec 1 radian/sec = 9.55 rev/min m is mass in kg r is radius of circle in meters V is the tangental velocity in m/s = ωr f is in Newtons Centripetal acceleration is a = Vdθ/dt = ω²r = Vω = V²/r V is the tangental velocity in m/s r is radius of circle in meters ω is angular velocity in radians/sec = v/r a is in m/s² Angular momentum in kg•m²/s L = Iω I is moment of inertia in kg•m² ω is angular velocity in radians/sec For a small particle moving in a circle: Angular momentum = ½ωmr² = rmV ω is angular velocity in radians/sec m is mass in kg V is the tangental velocity in m/s r is radius of circle in meters linear momentum converted to angular momentum for example a bullet hitting a door L = Pr where P = mV is linear momentum angular acceleration in rad/s² α = τ/I for constant α I = moment of inertia in kg•m² τ = torque in N•m τ = αI torque required for that accel. P = τω power to hold that speed with τ friction θ = θ₀ + ω₀t + ½αt² θ = θ₀ + ½(ω₀ + ω)t ω = ω₀ + αt ω² = ω₀² + 2α(θ – θ₀) θ = (Δω)(Δt)/2 ω = dθ/dt α = dω/dt α = Δω/Δt Tangental acceleration aᵢ = ΔVᵢ/Δt = αR Angular kinetic energy E in Joules E = ½Iω² ω is angular velocity in radians/sec I = moment of inertia in kg•m² moment of inertia A bodies moment of inertia about any given axis is the moment of inertia about the centroid plus the mass of the body times the distance between the point and the centroid squared. I₀ = Ic + md² I₀ is moment of inertia about point O Ic is moment of inertia about the centroid d is distance between the point and the centroid The moment of inertia of a point mass with respect to an axis is defined as the product of the mass times the distance from the axis squared. The moment of inertia of any extended object is built up from that basic definition. More that one point, they add. I is moment of inertia in kg•m² I = cMR² M is mass (kg), R is radius (meters) c = 1 for a ring or hollow cylinder c = 2/5 solid sphere around a diameter c = 7/5 solid sphere around a tangent c = ⅔ hollow sphere around a diameter c = ½ solid cylinder or disk around its center c = 1/4 solid cylinder or disk around a diameter c = 1/12 rod around its center, R = length c = ⅓ for a rod around its end, R = length c = 1 for a point mass M at a distance R from the axis of rotation c = 1/3 for a door, where r is the width for a wheel with r inner radius and R outer radius I = ½M(R²–r²) for a sphere on the end of a rod: Sphere radius R, mass M, rod length = L, rod mass m I = (2/5)MR² + M(L+R)² + (1/3)mL² Theorem of parallel axes: the moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes. cylinder or ball rolling at speed V linear KE = ½mV² rotational KE = ½Iω² = ½cmR²ω² = ½cmR²(V/R)² = ½cmV² Total KE = ½mV² + ½cmV² solid sphere total KE = ½mV² + (1/5)mV² = (7/10)mV² hollow sphere total KE = ½mV² + (1/3)mV² = (5/6)mV² solid cylinder total KE = ½mV² + (1/4)mV² = (3/4)mV² ring or hollow cylinder total KE = ½mV² + ½mV² = mV² cylinder or ball rolling down a slope PE at top results in both linear KE and rotational KE at the bottom. mgh = ½mV² + ½Iω² V = rω r is radius V² = (2gh) / (1 + I/mr²) for a solid cylinder that reduces to V² = (4/3)gh (1.33) for a solid sphere, that is V² = (10/7)gh (1.43) for a ring or hollow cylinder, that is V² = 2gh for a hollow sphere, that is V² = (6/5)gh (1.2) parallel axis theorem: It can be used to determine the second moment of area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular distance (r) between the axes. The moment of inertia about the new axis z is given by: Iz = Icm + mr² where: Icm is the moment of inertia of the object about an axis passing through its centre of mass; m is the object's mass; r is the perpendicular distance between the two axes. consider a body of mass m, moving in a circle of radius r, with an angular velocity of ω. speed is v = r·ω. centripetal (inward) acceleration is a = rω² = v²/r centripetal force is F = ma = rmω² = mv²/r momentum of the body is p = mv = rmω angular momentum is L = rmv = r²mω = I·ω kinetic energy is E = ½mv² = ½r²mω² = p²/2m = ½Iω = L²/2I Power (watts) = torque (N-m) x 2π x rotational speed (rev/sec) |
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