# Balancing

 Question 1
A rigid body in the X-Y plane consists of two point masses (1 kg each) attached to the ends of two massless rods, each of 1 cm length, as shown in the figure. It rotates at 30 RPM counter-clockwise about the Z-axis passing through point O. A point mass of $\sqrt{2}$ kg, attached to one end of a third massless rod, is used for balancing the body by attaching the free end of the rod to point O. The length of the third rod is ________ cm.

 A $1$ B $\sqrt{2}$ C $\frac{1}{\sqrt{2}}$ D $\frac{1}{2\sqrt{2}}$
GATE ME 2022 SET-2   Theory of Machine
Question 1 Explanation:

$m_1=1kg, m_2=1kg, r_1=1cm, r_2=1cm$
Balancing mass, $m_b=\sqrt{2}kg$
Making force polygon for complete balance

From the right angle triangle,
\begin{aligned} (m_br_b)^2&=(m_1r_1)^2+(m_2r_2)^2 \\ (\sqrt{2}r_b)^2&=(1\times 1)^2+(1\times 1)^2 \\ 2r_b^2&=2 \\ r_b&=1 \end{aligned}
 Question 2
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 2 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 3
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 3 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 4
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 4 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 5
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 5 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 6
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 6 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 6 questions to complete.