A particle is at the origin of the \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\) axis in the standard coordinate system in \(\mathrm{R}^{3}\). The particle is subject to the following forces: \(\mathrm{F}_{1}\) : force of 5 units acting along \(0 \mathrm{x}\) \(\mathrm{F}_{2}\) : force of 3 units acting along \(z 0\) \(\mathrm{F}_{3}:\) force of 2 units acting along \(0 \mathrm{y}\) \(\mathrm{F}_{4}\) : force of $2 \sqrt{2}\( units acting towards 0 at an angle of \)\pi / 4\( to the \)\mathrm{x}$ and \(\mathrm{y}\) axes in the xy plane. Find the resultant force on the particle.

Use vector addition in \(R^{2}\) to compute the actual speed and direction of an airplane subject to wind conditions. i) Suppose the plane is flying \(200 \mathrm{mi} / \mathrm{hr}\). due north with a tailwind which is \(50 \mathrm{mi} / \mathrm{hr}\). north. ii) The plane is flying \(120 \mathrm{mi} / \mathrm{hr}\). north and the wind has velocity \(50 \mathrm{mi} / \mathrm{hr}\). east.

Three particles of mass \(\mathrm{m}\) are spaced at distances \(\ell\) on a light vertical string which hangs freely from a point of support. All subsequent displacements are in a fixed vertical plane. How do the particles behave if: i) they are displaced by constant horizontal forces of magnitudes \(\mathrm{f}_{1}, \mathrm{f}_{2}\) and \(\mathrm{f}_{3} ;\) (ii) they are subject to horizontal forces with magnitudes $\mathrm{p}_{1} \sin \mathrm{kt}, \mathrm{p}_{2} \sin \mathrm{kt}\( and \)\mathrm{p}_{3} \sin \mathrm{kt} ;$ (iii) they oscillate freely?

Three particles each of unit mass, are equally spaced at distances \(\ell\) on a light horizontal string fastened rigidly at the ends and resting on a smooth horizontal table. What happens to the particles if i) they are subject to constant horizontal forces \(f_{1}, f_{2}, f_{3} ;\) ii) they are subject to periodic horizontal forces \(\mathrm{p}_{1}\) sin \(\mathrm{kt}, \mathrm{p}_{2}\) $\sin \mathrm{kt}, \mathrm{p}_{3} \sin \mathrm{kt}$ iii) they are allowed to oscillate freely in the plane.

Determine the number of dimensionless groups in a problem of transient heat conduction in a rod with initial temperature \(\mathrm{T}_{\mathrm{i}}\) and end temperature \(\mathrm{T}_{0}\).

i) Find the work done by moving an object along the vector $2 \mathrm{i}+6 \mathrm{j}\( if the force acting on the particle is \)\mathrm{i}+2 \mathrm{j}$. ii) A uniform bar \(6 \mathrm{ft}\). by 40 lbs rotates about a fixed horizontal axis at 0 . Compute the work done by the weight of the bar as the bar rotates from the position vector \(3 \mathrm{i} \mathrm{ft}\). to the position vector \(-1.8 \mathrm{i}-2.4 \mathrm{j} \mathrm{ft}\). where the position vector of the bar is the vector from the origin to the center of mass of the bar.