What are the three principle kinematic relations that may be used to completely describe the motion of a particle translating with variable acceleration?
1) v = ds/dt, where v denotes the particle's velocity, ds is the change in position (i.e. the displacement), and dt is the differential of time elapsed in displacing the particle that amount. 2) a = dv/dt = d^2s/dt^2, where dv is the change in the particle's velocity across some differential of time dt. Note that the second equation is equivalent due to Equation (1) above. 3) ads = vdv, eliminating dt as a variable, acceleration may be directly related to the particles change in position (i.e. the displacement), and the change in velocity dv. Note all three equations are vector equations having both magnitude and direction. Thus, they each yield 3 scalar equations that may be expressed in terms of x,y, and z given the selection of a Cartesian cooridinate system to procede with the analysis.
Some torque T is induced on a shaft such that it's subjected to pure torsion. Answer the following questions: (a) What is "Torsion"? What is its relation to Torque? (b) What assumptions should be made about the material and the geometry of the shaft when computing torsion? (c) What parameters are needed to compute torsion? (d) For some given torque T, would a hollow shaft or a solid shaft experience greater torsion?
(a) Torsion occurs when a pair of forces of equal magnitude & opposite direction act on a given body (e.g. shafts) such that the induced loading twists the body. Torsion is also known as the "twisting moment" or the "torsional moment." The torque T is equal to the product of the force applied and the distance between the point of application of the force F and the axis of the shaft. (b) 1. The material is homogenous and isotropic. 2. The stresses are within elastic limit. 3. C/S which are plane before applying twisting moment remain plane even after the application of twisting moment. This is known as the Elastic Bending Theory. 4. Radial lines remain radial even after applying torsional moment.5. The twist along the shaft is uniform. (c) The equation describing the torsion developed in the shaft is given by: T/J=Cθ/L=q/R Where, T: Torque J: Polar Moment of Inertia C: Modulus of Rigidity L: Shaft Length q: Shear Stress R: Shaft Radius (d) The torque transmitted by the hollow shaft is greater than the solid shaft. For same material, length and given torque, the weight of the hollow shaft will be less compared to solid shaft.
For the following four functions given in Cartesian (xyz) coordinates in Euclidean space, specify the quadratic surfaces defined across the open interval. After classifying the conical surfaces, sketch the graphs of each of the four functions. Include all 3 principal axes and any relevant points of reference in your sketch, (a) x2 −z2 = 0 (b) x2 + 2y2 + 4z2 = x + 2y + 4z (c) x2 + y2 −z = 1 (d) 4x2 + y2 −z2 = 2y−2z
(a) This equation describes 2 planes intersecting along the y-axis. Note that there are an infinite amount of solutions, as x and y may take on any real value. (b) This equation describes an ellipsoid, which can be seen by rearranging the equation in the form: [(x−1/2)/2] + [2(y −1/2)]/2 + [4(z −1/2)]/2 = 7/4. Note the center C of the ellipsoid is located at the coordinate triplet (1/2,1/2,1/2). (c) This equation describes a paraboloid with its concave side opening along the positive z-axis. The vertex of this paraboloid is given by the coordinate triplet (0,0,−1). (d) This equation describes an elliptical cone with its concave side opening along the positive z-axis. Rearranging the equation in the form: 4x2 + (y − 1)2 = (z − 1)2 , shows that the vertex is coincident with the coordinate triplet (0,1,1).