Calculus - Engineering Mathematics - Petroleum Engineering Previous Year Questions

A uniform cantilever beam \(ABC\) of length \(L\) is subjected to a point load \(P\) at point \(B\) and a concentrated moment \(M\) at point \(C\) (as shown in figure). Let \(E\) be the Young’s modulus of the beam material and \(I\) be the area moment of inertia of the beam’s cross-section. Assuming the validity of the Euler-Bernoulli theory of slender beams, the downward deflection at point \(C\) is

True centrifugal casting process in horizontal configuration is to be used for casting a metallic cylinder with outside diameter 0.275 m and inside diameter 0.250 m. If G-factor (ratio of centrifugal force experienced by the rotating cast metal to its weight) is 65 and acceleration due to gravity is 9.8 m/s², the minimum rotational speed (in rpm) required is closest to

Let \(\vec{F}(x,y) = e^y \sin x \, \hat{i} + x^2 \, \hat{j}\) for \((x,y) \in \mathbb{R}^2\). If \(C\) is the circle \(x^2 + y^2 = 4\) oriented in the anticlockwise direction then \(\oint_C \vec{F} \cdot d\vec{R}\) equals

\(\mathbf{r}(t) = \frac{\sin 3t}{t}\mathbf{i} + (t+2)^t\mathbf{j} + (t+1)\frac{\sin t}{t}\mathbf{k}\), with \(i, j,\) and \(k\) being the unit vectors along \(x, y,\) and \(z\) directions, respectively.
The value of \(\lim_{t \to 0} \mathbf{r}(t)\) is __________.