# Bearings, Shafts and keys

 Question 1
A shaft AC rotating at a constant speed carries a thin pulley of radius $r = 0.4 m$ at the end C which drives a belt. A motor is coupled at the end A of the shaft such that it applies a torque $M_z$ about the shaft axis without causing any bending moment. The shaft is mounted on narrow frictionless bearings at A and B where $AB = BC = L = 0.5 m$. The taut and slack side tensions of the belt are $T_1= 300 N$ and $T2 = 100 N$, respectively. The allowable shear stress for the shaft material is 80 MPa. The selfweights of the pulley and the shaft are negligible. Use the value of $\pi$ available in the on-screen virtual calculator. Neglecting shock and fatigue loading and assuming maximum shear stress theory, the minimum required shaft diameter is _______ mm (round off to 2 decimal places). A 45.32 B 32.78 C 23.94 D 18.25
GATE ME 2022 SET-2   Machine Design
Question 1 Explanation: \begin{aligned} M_{max}&=400 \times L =400 \times 0.5 \times 10^3\\ &=200 \times 10^3 \; N.mm\\ T_{max}&=M_z\\ &=(T_1-T_1) \times r\\ &=(200 \times 0.4) \times 10^3\\ T_{max}&=80 \times 10^3 \; N.mm \end{aligned}

Critical particle :
According to maximum shear stress theory,
\begin{aligned} \frac{16}{\pi d^3}\sqrt{M_{max}^2+T_{max}^2}&=\frac{S_{{ys}}}{FOS}\\ \frac{16 \times 10^3}{\pi d^3}\sqrt{200^2+80^2}&=\frac{80}{1}\\ \Rightarrow d&=23.94mm \end{aligned}
 Question 2
A shaft of length $L$ is made of two materials, one in the inner core and the other in the outer rim, and the two are perfectly joined together (no slip at the interface) along the entire length of the shaft. The diameter of the inner core is $d_i$ and the external diameter of the rim is $d_o$, as shown in the figure. The modulus of rigidity of the core and rim materials are $G_i$ and $G_o$, respectively. It is given that do $d_o=2d_i$ and $G_i=3G_o$. When the shaft is twisted by application of a torque along the shaft axis, the maximum shear stress developed in the outer rim and the inner core turn out to be $\tau _o$ and $\tau _i$, respectively. All the deformations are in the elastic range and stress strain relations are linear. Then the ratio $\tau _i /\tau_o$ is ______ (round off to 2 decimal places). A 1.15 B 2.65 C 1.85 D 1.5
GATE ME 2022 SET-2   Machine Design
Question 2 Explanation:
Given $G_i=3G_o, d_o=2d_i, l_i=l_o,\theta _i=\theta _o \text{ (It is a rigid joint)}$
Find $\frac{T_i}{T_o}=?$
\begin{aligned} \frac{T}{J}&=\frac{\tau }{r}=\frac{G\theta }{L}\\ \tau _{max}(\text{in core})&=\frac{G\theta \times r_{max}}{L}\\ &=\frac{G_i \times \theta _i \times \frac{d_i}{2}}{L_i}=\tau _i\\ \tau _{max}(\text{rim})&=\frac{G_o \times \theta _o\times \frac{d_o}{2} }{L_o}=\tau _o\\ \frac{T_i}{\tau _o}&=\frac{G_i \times \theta _i \times \frac{d_i}{2}}{L_i} \times \frac{L_o}{G_o \times \theta _o\times \frac{d_o}{2} }\\ \frac{T_i}{T_o}&=\frac{G_i}{G_o} \times \frac{d_i}{d_o}=3 \times \frac{1}{2}=1.5 \end{aligned}

 Question 3
A short shoe external drum brake is shown in the figure. The diameter of the brake drum is 500 mm. The dimensions a= 1000 mm, b = 500 mm and c = 200 mm. The coefficient of friction between the drum and the shoe is 0.35. The force applied on the lever F= 100 N as shown in the figure. The drum is rotating anti-clockwise. The braking torque on the drum is______ N.m (round off to two decimal places). A 12.75 B 8.55 C 20.35 D 26.85
GATE ME 2019 SET-2   Machine Design
Question 3 Explanation:
$\mathrm{F}_{\mathrm{s}}=\mu_{\mathrm{s}} \mathrm{N}=0.35 \times \mathrm{N}$ Taking summation of moment about O = zero
$\begin{array}{l} \Sigma M_{0}=0 \\ N \times 500-F_{s} \times 200-F \times 1000=0 \\ N \times 500-0.35 \times N \times 200-100 \times 1000=D \\ N=\frac{100 \times 1000}{(500-70)}=232.558 \mathrm{N} \end{array}$
With respect to FBD of drum
$\text { Braking torque } \mathrm{M}=\mathrm{F}_{\mathrm{s}} \times \mathrm{r}=\frac{0.35 \times 232.558 \times 500}{2 \times 1000}$
$\mathrm{M}=20.35 \mathrm{N}-\mathrm{m}$
 Question 4
A self-aligning ball bearing has a basic dynamic load rating ( $C_{10}$ for $10^{6}$ revolutions) of 35 kN. If the equivalent radial load on the bearing is 45 kN, the excepted life (in revolutions) is
 A below 0.5 B 0.5 to 0.8 C 0.8 to 1.0 D above 1.0
GATE ME 2018 SET-1   Machine Design
Question 4 Explanation:
\begin{aligned} C &=35 \mathrm{kN} \\ \mathrm{P}_{C} &=45 \mathrm{kN} \\ L_{90} &=\left(\frac{C}{P_{C}}\right)^{3}=\left(\frac{35}{45}\right)^{3}=0.4705 \mathrm{MR} \end{aligned}
 Question 5
Which of the bearings given below SHOULD NOT be subjected to a thrust load?
 A Deep groove ball bearing B Angular contact ball bearing C Cylindrical (straight) roller bearing D Single row tapered roller bearing
GATE ME 2016 SET-3   Machine Design
Question 5 Explanation:
Correct option is (C)

There are 5 questions to complete.

### 1 thought on “Bearings, Shafts and keys”

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