# Planar Mechanisms

 Question 1
In the configuration of the planar four-bar mechanism at a certain instant as shown in the figure, the angular velocity of the 2 cm long link is $\omega _2 =5$rad/s. Given the dimensions as shown, the magnitude of the angular velocity $\omega _4$ of the 4 cm long link is given by _____ rad/s (round off to 2 decimal places).

 A 1.25 B 0.75 C 2.25 D 3.75
GATE ME 2022 SET-2   Theory of Machine
Question 1 Explanation:

Number of I-centers $4c_2=6$ $\omega _2=5 \; rad/sec$
$\omega _4=?$
For
\begin{aligned} V_{I_{24}}&=I_{12}I_{24}\omega _{2}=I_{14}I_{24}\omega_4\\ &=2 \times 5 =8 \times \omega_4\\ \omega_4&=1.25 \; rad/sec \end{aligned}
 Question 2
A planar four-bar linkage mechanism with 3 revolute kinematic pairs and 1 prismatic kinematic pair is shown in the figure, where AB$\perp$CE and FD$\perp$ CE. The T-shaped link CDEF is constructed such that the slider B can cross the point D, and CE is sufficiently long. For the given lengths as shown, the mechanism is

 A a Grashof chain with links AG, AB, and CDEF completely rotatable about the ground link FG B a non-Grashof chain with all oscillating links C a Grashof chain with AB completely rotatable about the ground link FG, and oscillatory links AG and CDEF D on the border of Grashof and non-Grashof chains with uncertain configuration(s)
GATE ME 2022 SET-1   Theory of Machine
Question 2 Explanation:
The given mechanism is

As we know sliding pair is a special. Case of turning pair with infinite lengths limit. So the equivalent diagram would be. Since two parallel lines meets at infinite point $O, 2O_2$ are same.

$l_1=3 cm$ shortest line,
$l_2=5cm$
$l_3=l_{3(\infty )}=L_x+3 ,$ longest link
$l_4=l_{4(\infty )}=L_x+1.5$
For Grashoff's rule to satisfy
$l_1+l_3 \leq l_2 +l_4$
$3+L_x+3 \leq 5+L_x+1.5$
$6 \leq 6.5$
LHS is less than RHS.
Hence, Grashoff's rule is satisfied in this mechanism. Since shortest link is fixed. It will be a double crank mechanism.

 Question 3
A rigid triangular body, PQR, with sides of equal length of 1 unit moves on a flat plane. At the instant shown, edge QR is parallel to the x-axis, and the body moves such that velocities of points P and R are $V_P \; and \; V_R$, in the x and y directions, respectively. The magnitude of the angular velocity of the body is
 A $2V_R$ B $2V_P$ C $V_R/\sqrt{3}$ D $V_P/\sqrt{3}$
GATE ME 2019 SET-2   Theory of Machine
Question 3 Explanation:
$\begin{array}{l} \Rightarrow \mathrm{V}_{\mathrm{R}}=(\mathrm{IR}) \omega \\ \Rightarrow \omega=\frac{\mathrm{V}_{\mathrm{R}}}{(\mathrm{IR})} \\ \Rightarrow \omega \times \frac{\mathrm{V}_{\mathrm{R}}}{\frac{1}{2}} \\ \Rightarrow \omega=2 \mathrm{V}_{\mathrm{R}} \end{array}$

 Question 4
In a four bar planar mechanism shown in the figure, AB = 5 cm, AD = 4 cm and DC = 2 cm. In the configuration shown, both AB and DC are perpendicular to AD. The bar AB rotates with an angular velocity of 10 rad/s. The magnitude of angular velocity (in rad/s) of bar DC at this instant is
 A 0 B 10 C 15 D 25
GATE ME 2019 SET-1   Theory of Machine
Question 4 Explanation:
$\begin{array}{l} \mathrm{AB}=5 \mathrm{cm} \\ \mathrm{AD}=4 \mathrm{cm} \\ \mathrm{DC}=2 \mathrm{cm} \\ \omega_{\mathrm{AB}}=10 \mathrm{rad} / \mathrm{s} \\ \because A \mathrm{B} \| \mathrm{DC} \\ \therefore A B \cdot \omega_{\mathrm{AB}}=\mathrm{DC} \cdot \omega_{\mathrm{DC}} \\ 5 \times 10=2 \times \omega_{\mathrm{DC}} \\ \omega_{\mathrm{DC}}=25 \mathrm{rad} / \mathrm{s} \end{array}$
 Question 5
In a slider-crank mechanism, the lengths of the crank and the connecting rod are 100mm and 160mm, respectively. The crank is rotating with an angular of 10 radian/s counter-clockwise. The magnitude of linear velocity (in m/s) of the piston at the instant corresponding to the configuration shown in the figure is_____.
 A 2 B 1.5 C 1 D 0.5
GATE ME 2017 SET-2   Theory of Machine
Question 5 Explanation:

After plotting I-centres

Here, $I_{23}$ and $I_{24}$ will come at same point
Applying angular Velocity Theorem at $I_{24}$
\begin{aligned} \therefore \quad \omega_{2}\left(I_{24} I_{12}\right) &=V_{4}=V_{B} \\ V_{B} &=\omega_{2}\left(I_{24} I_{12}\right)=10 \times 0.1 \\ \left(I_{24} I_{12}=100 \mathrm{mm}\right.&=0.1 \mathrm{m}=A B) \\ V_{B} &=1 \mathrm{m} / \mathrm{s} \end{aligned}

There are 5 questions to complete.

### 5 thoughts on “Planar Mechanisms”

1. I think questions 1 diagram are not matchable

• Thank you NAVEENKUMAR,
We have updated the figure.