Balancing

Question 1
Consider a reciprocating engine with crank radius R and connecting rod of length L. The secondary unbalance force for this case is equivalent to primary unbalance force due to a virtual crank of _____
A
radius \frac{L^2}{4R} rotating at half the engine speed
B
radius \frac{R}{4} rotating at half the engine speed
C
radius \frac{R^2}{4L} rotating at twice the engine speed
D
radius \frac{L}{2} rotating at twice the engine speed
GATE ME 2021 SET-1   Theory of Machine
Question 1 Explanation: 
Unbalanced secondary force,
F_s=mr\omega \frac{\cos 2\theta }{n}=m \times \left ( \frac{R}{4n} \right )\times (2\omega )^2 \times \cos 2\theta
Balancing radius =\frac{R}{4n}=\frac{R}{4 \times \frac{L }{R}}=\frac{R^2}{4L }
Balancing crank speed =2\omega
Question 2
Two masses A and B having mass m_a \; and \; m_b, respectively, lying in the plane of the figure shown, are rigidly attached to a shaft which revolves about an axis through O perpendicular to the plane of the figure. The radii of rotation of the masses m_a \; and \; m_b are r_a \; and \; r_b, respectively. The angle between lines OA and OB is 90^{\circ}. If m_a =10 kg, m_b=20kg, r_a=200mm, r_b=400mm, then the balance mass to be placed at a radius of 200 mm is ______ kg (round off to two decimal places).
A
21.25
B
28.56
C
35.55
D
41.23
GATE ME 2019 SET-2   Theory of Machine
Question 2 Explanation: 
\begin{array}{l} \mathrm{m}_{\mathrm{A}}=10 \mathrm{kg} \\ \mathrm{m}_{\mathrm{B}}=20 \mathrm{kg} \\ \mathrm{r}_{\mathrm{A}}=200 \mathrm{mm} \\ \mathrm{r}_{\mathrm{B}}=400 \mathrm{mm} \\ \mathrm{r}=200 \mathrm{mm} \\ \mathrm{mr} \omega^{2}=\sqrt{\left(\mathrm{m}_{\mathrm{A}} \mathrm{r}_{\mathrm{A}} \omega^{2}\right)^{2}+\left(\mathrm{m}_{\mathrm{B}} \mathrm{r}_{\mathrm{B}} \omega^{2}\right)^{2}} \\ \Rightarrow \mathrm{m} \times 200=\sqrt{(10 \times 200)^{2}+(20 \times 400)^{2}} \\ \Rightarrow \mathrm{m}=41.23 \mathrm{kg} \end{array}
Question 3
Three masses are connected to a rotating shaft supported on bearings A and B as shown in the figure. The system is in a space where the gravitational effect is absent. Neglect the mass of shaft and rods connecting the masses.for m_{1}=10 kg,m_{2}=5 kg and m_{3}=2.5 kg and for a shaft angular speed of 1000 radian/s, the magnitude of the bearing reaction (in N) at location B is_____
A
1
B
2
C
3
D
0
GATE ME 2017 SET-2   Theory of Machine
Question 3 Explanation: 


It means all three masses are in same plane
let us calculates the net force
\begin{array}{c} \Sigma F_{x}= \Sigma \mathrm{m} \mathrm{r} \omega^{2} \cos \theta \\ =\left(10 \times 0.1 \times \omega^{2}\right)-\left(5 \times 0.2 \times 6^{2} \cos 60^{\circ}\right) \\ -\left(2.5 \times 0.4 \times \omega^{2} \cos 60^{\circ}\right) \\ =\left[1-\left(5 \times 0.2 \times \frac{1}{2}\right)-\left(2.5 \times 0.4 \times \frac{1}{2}\right)\right] \omega^{2}=0 \\ \Sigma F_{y}= \Sigma \mathrm{mr} \omega^{2} \sin \theta \\ =\left(5 \times 0.2 \times \omega^{2} \sin 60^{\circ}\right) \\ -\left(2.5 \times 0.4 \times \omega^{2} \sin 60^{\circ}\right) \\ =\left[\left(1 \times \frac{\sqrt{3}}{2}\right)-\left(1 \times \frac{\sqrt{3}}{2}\right)\right] \omega^{2} \\ =\left[\left(\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2}\right)\right] \omega^{2}=0 \end{array}
Net force,
=\sqrt{\Sigma F_{x}^{2}+\Sigma F_{y}^{2}}=0
Therefore reaction at B is zero.
R_{B}=0
Question 4
Two masses m are attached to opposite sides of a rigid rotating shaft in the vertical plane. Another pair of equal masses m1 is attached to the opposite sides of the shaft in the vertical plane as shown in figure. Consider m = 1 kg, e = 50 mm, e1 = 20 mm, b = 0.3 m, a = 2 m and a1 = 2.5 m. For the system to be dynamically balanced, m1 should be ________ kg.
A
1kg
B
2kg
C
3kg
D
4kg
GATE ME 2016 SET-3   Theory of Machine
Question 4 Explanation: 


Balance moment of all forces
\begin{array}{c} \omega^{2}\left(m_{1} \times 0.02 \times 2.5\right)+(1 \times 0.05 \times 0.3) \omega^{2}=\left(m_{1} \times\right. \\ 0.02 \times 0) \omega^{2}+(1 \times 0.05 \times 2.3) \omega^{2} \\ m_{1}=2 \mathrm{kg} \end{array}
Question 5
A cantilever type gate hinged at Q is shown in the figure. P and R are the centers of gravity of the cantilever part and the counterweight respectively. The mass of the cantilever part is 75 kg. The mass of the counterweight, for static balance, is
A
75kg
B
150kg
C
225kg
D
300kg
GATE ME 2008   Theory of Machine
Question 5 Explanation: 
Let counterweight at R is m kg

For static condition
\begin{aligned} \Sigma M_{Q} &=0 \\ m \times 0.5 &=75 \times 2 \\ m &=300 \mathrm{kg} \end{aligned}
There are 5 questions to complete.

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