GATE Mechanical Engineering 2020 SET-2

\begin{aligned} X_{1} &\sim N\left(\mu_{1}, \sigma_{1}\right)\\ \text{and }\quad X_{2} &\sim N\left(\mu_{2}, \sigma_{2}\right)\\ \text{then }\quad X_{1}+X_{2} &\sim N\left(\mu_{1}+\mu_{2}, \sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}\right) \end{aligned}
Always normally distributed.

\begin{array}{l} S=\frac{P+P^{T}}{2} \\ V=\frac{P-P^{T}}{2} \\ P=S+V=\left(\begin{array}{ccc} -4 & 2 & 5 \\ 6 & 3 & 7 \\ -1 & 0 & 2 \end{array}\right) \end{array}

I=\int_{0}^{1} \int_{0}^{x^{2}} x y^{2} d y d x

Change on rules, I=\int_{y=0}^{1} \int_{x=\sqrt{y}}^{1} x y^{2} d x d y

\begin{aligned} y^{\prime\prime}-y &=1 \\ y(0) &=1 \\ y^{\prime}(0) &=1 \\ L\left\{y^{\prime\prime}-y\right\} &=L\{1\} \\ s^{2} Y(s)-s y(0)-y^{\prime}(0)-y(s) &=\frac{1}{s} \\ y(s) &=\frac{1}{s\left(s^{2}-1\right)}\\ & =\frac{1}{s(s+1)(s-1)} \end{aligned}

Weigh = W
Note:
(i) When the cylinder is about to make out of the curb, it will loose its contact at point A, only contact will be at it B.
(ii) At verge of moving out of curb, Roller will be in equation under W, F and contact force from B and these three forces has to be concurrent so contact force from B will pass through C.
(iii) Even the surfaces are rough but there will be no friction at B for the said condition.
FBD

For pure rolling


\begin{array}{l} v_{P}=v=(P R) \omega \quad \ldots(i)\\ v_{Q}=2 v=(Q R) \omega \quad \ldots(ii) \end{array}
Divide by (ii) to (i),
2=\frac{Q R}{P R} \Rightarrow Q R=2(P R)
\begin{aligned} P R+Q R &=2 r \\ P R+2(P R) &=2 r \\ P R &=\frac{2}{3} r \\ \text{From equation(i) }\quad v &=\left(\frac{2}{3} r\right) \omega \Rightarrow \omega=\frac{3 v}{2 r} \end{aligned}

\frac{d^{2} x}{d t^{2}}+3\left(\frac{d x}{d t}\right)+9 x=10 \sin 5 t
Comparing with standard equation:
\ddot{x}+\left(2 \xi \omega_{n}\right) \dot{x}+\left(\omega_{n}^{2}\right) x=\left(\frac{F_{o}}{m}\right) \sin \omega t
\begin{aligned} 2 \xi \omega_{n} &=3 \\ 2 \xi \times 3 &=3 &\left[\begin{array}{l} \omega_{n}^{2}=9 \\ \omega_{n}=3 \end{array}\right] \\ \xi &=\frac{1}{2}=0.5 \end{aligned}

They are:
1. Double crank mechanism
2. Crank-rocker mechanism
3. Double rocker mechanism

In Ultrasonic machining, the function of horn (also called concentrator, it is a tapered metal bar) is to amplify and focus vibration of the transducer to an adequate intensity for driving the tool to fulfll the cutting operation.

Processes with low rate of heat input (slow heating) tend to produce high total heat constant within the metal, slow cooling rates, and large heat-affected zones. high heat input process, have low total heats, fast cooling rates and small heat affected zones