# Brakes and Clutches

 Question 1
The braking system shown in the figure uses a belt to slow down a pulley rotating in the clockwise direction by the application of a force $P$. The belt wraps around the pulley over an angle $\alpha = 270$ degrees. The coefficient of friction between the belt and the pulley is 0.3. The influence of centrifugal forces on the belt is negligible.
During braking, the ratio of the tensions $T_1$ to $T_2$ in the belt is equal to __________. (Rounded off to two decimal places)
Take $\pi =3.14$. A 2.12 B 6.25 C 4.11 D 8.25
GATE ME 2023   Machine Design
Question 1 Explanation:
\begin{aligned} \mu & =0.3 \\ \alpha & =270^{\circ}=\frac{3 \pi}{2} \\ \frac{T_{1}}{T_{2}} & =\mathrm{e}^{\mu \alpha} \end{aligned}
So, $\quad \frac{T_{1}}{T_{2}}=e^{0.3 \times \frac{3 \pi}{2}}=4.11$
 Question 2
A short shoe drum (radius 260 mm) brake is shown in the figure. A force of 1 kN is applied to the lever. The coefficient of friction is 0.4. The magnitude of the torque applied by the brake is ________N.m (round off to one decimal place).
 A 150 B 175 C 200 D 250
GATE ME 2021 SET-1   Machine Design
Question 2 Explanation: \begin{aligned} R_{N}(500)+F_{\gamma}[310-260] &-1000 \times 1000=0 \\ R_{N}(500)+0.4\left(R_{N}\right)(50) &-1000 \times 1000=0 \\ R_{N} &=1923.076 \mathrm{~N} \\ F_{r} &=\mu R_{N}=769.23 \mathrm{~N} \\ T_{f} &=F_{r} \times R=200 \mathrm{~N}-\mathrm{m} \end{aligned}

 Question 3
A helical spring has spring constant k. If the wire diameter, spring diameter and the number of coils are all doubled then the spring constant of the new spring becomes
 A k/2 B k C 8k D 16k
GATE ME 2020 SET-2   Machine Design
Question 3 Explanation:
\begin{aligned} k_{(\text {sping })} &=\frac{G d^{4}}{8 D^{3} n} \\ k_{\text {new }} &=\frac{G(2 d)^{4}}{8(2 D)^{3}(2 n)}=\frac{G d^{4}}{8 D^{3} n} \\ \text{Hence}\qquad k_{\text {new }} &=k \end{aligned}
 Question 4
In a disc-type axial clutch, the frictional contact takes place within an annular region with outer and inner diameters 250 mm and 50 mm, respectively. An axial force $F_1$ is needed to transmit a torque by a new clutch. However, to transmit the same torque, one needs an axial force $F_2$ when the clutch wears out. If contact pressure remains uniform during operation of a new clutch while the wear is assumed to be uniform for an old clutch and the coefficient of friction does not change, then the ratio $F_1/F_2$ is_________ (round off to 2 decimal places).
 A 0.22 B 0.46 C 0.87 D 0.93
GATE ME 2020 SET-1   Machine Design
Question 4 Explanation:
\begin{aligned} T &=\mu W \times R_{m} \\ T_{\text {new }} &=\mu \times W \times R_{\text {new }} \\ T_{\text {new }} &=\mu \times W_{\text {new }} \times \frac{2}{3} \frac{\left(R_{0}^{3}-R_{i}^{3}\right)}{R_{0}^{2}-R_{i}^{2}} \\ T_{\text {odd }} &=\mu \times W \times R_{\text {old }} \\ T_{\text {new }} &=T_{\text {dd }} \\ W_{\text {new }} \times \frac{2}{3} \frac{\left(R_{0}^{3}-R_{i}^{3}\right)}{R_{0}^{2}-R_{i}^{2}} &=W_{\text {old }} \times \frac{\left(R_{0}+R_{i}\right)}{2} \\ \frac{W_{\text {new }}}{W_{\text {old }}} &=0.871 \end{aligned}
 Question 5
A single block brake with a short shoe and torque capacity of 250 $N\cdot m$ is shown. The cylindrical brake drum rotates anticlockwise at 100 rpm and the coefficient of friction is 0.25. The value of a, in mm (round off to one decimal place), such that the maximum actuating force P is 2000 N, is ________ A 212.5 B 256.4 C 159.8 D 753.5
GATE ME 2019 SET-1   Machine Design
Question 5 Explanation: $\mu=0.25$
76 Torque $=250 \mathrm{Nm}$
$\mu \mathrm{F} \times \mathrm{a}=250$
$\Rightarrow 0.25 \times \mathrm{F} \times \mathrm{a}=250$
$\Rightarrow \mathrm{F} \times \mathrm{a}=1000$
In FBD of block and lever, $\Sigma \mathrm{M}_{\text {fullcrum }}=0$
$\mathrm{F} \times \mathrm{a}+\mu \mathrm{F} \times \frac{\mathrm{a}}{4}=\mathrm{P} \times 2.5 \mathrm{a}$
$F \times a\left(1+\frac{\mu}{4}\right)=P \times 2.5$
$1000\left(1+\frac{0.25}{4}\right)=2000 \times 2.5 \mathrm{a}$
$\Rightarrow \mathrm{a}=0.2125 \mathrm{m}=212.5 \mathrm{mm}$

There are 5 questions to complete.