Friction


Question 1
A cylindrical disc of mass m=1 \; kg and radius r=0.15\;m was spinning at \omega =5 \; rad/s when it was placed on a flat horizontal surface and released (refer to the figure). Gravity g acts vertically downwards as shown in the figure. The coefficient of friction between the disc and the surface is finite and positive. Disregarding any other dissipation except that due to friction between the disc and the surface, the horizontal velocity of the center of the disc, when it starts rolling without slipping, will be _________ m/s (round off to 2 decimal places).

A
0
B
0.25
C
0.5
D
0.75
GATE ME 2022 SET-1   Engineering Mechanics
Question 1 Explanation: 


About point P there is no external torque.
[Because Torques of N and 'mg' balances and friction force pass through point P, hence its torque is zero]
\therefore \left [ \frac{d\vec{L}}{dt} =\vec{\tau }_{ext}\right ]_{\text{about P}}=0
where \vec{\tau }= Torque and \vec{L }= Angular momentum
\begin{aligned} \vec{L_i}&=\vec{L_f}\\ I_0\omega &=I_0\omega '+mvr\\ \frac{mr^2}{2}\omega &=\frac{mr^2}{2}\omega'+m\omega 'r^2\\ \therefore \omega '&=\frac{\omega }{3}=\frac{5}{3}\\ \Rightarrow v&=\omega ' \times r\\&=\frac{5}{3} \times 0.15=0.25 m/s \end{aligned}
Question 2
Two rigid massless rods PR and RQ are joined at frictionless pin-joint R and are resting on ground at P and Q, respectively, as shown in the figure. A vertical force F acts on the pin R as shown. When the included angle \theta \lt 90^{\circ} , the rods remain in static equilibrium due to Coulomb friction between the rods and ground at locations P and Q. At \theta = 90^{\circ} , impending slip occurs simultaneously at points P and Q. Then the ratio of the coefficient of friction at Q to that at P P(\mu _Q/\mu _P) is _________ (round off to two decimal places).

A
5.76
B
9.32
C
4.25
D
8.65
GATE ME 2022 SET-1   Engineering Mechanics
Question 2 Explanation: 


By pythagoras theorem:
PQ = 13 m
By Geometry:
\begin{aligned} 5^2-PR^{'2}&=10^2-AR^{'2}\\ QR'&=13-PR'\\ \Rightarrow PR'&=\frac{25}{13}m\\ \Rightarrow QR'&=\frac{144}{13}m\\ \Sigma M_P&=0\\ \Rightarrow F(PR')&=N_Q(PQ)\\ \Rightarrow N_Q&=\frac{25}{169}F\\ \Sigma F&=0\\ \Rightarrow N_P&=F-\frac{25}{169}F\\ &=\frac{144}{169}F \end{aligned}
Now, f_P=\mu _PN_P, f_Q=\mu _QN_Q, \Sigma F_x=0
\begin{aligned} \mu _PN_P&=\mu _QN_Q\\ \frac{\mu _Q}{\mu _P}&=\frac{N_P}{N_Q}\\ &=\frac{144}{25}=5.76 \end{aligned}


Question 3
A block of negligible mass rests on a surface that is inclined at 30^{\circ} to the horizontal plane as shown in the figure. When a vertical force of 900 N and a horizontal force of 750 N are applied, the block is just about to slide.

The coefficient of static friction between the block and surface is _____ (round off to two decimal places).
A
0.28
B
0.36
C
0.17
D
0.05
GATE ME 2021 SET-2   Engineering Mechanics
Question 3 Explanation: 

After forces are applied block is just about move (mass is negligible). Calculate coefficient of friction
\begin{aligned} F_{H} &=750 \mathrm{~N} \\ F_{V} &=900 \mathrm{~N} \\ \theta &=30^{\circ} \end{aligned}


\begin{aligned} N &=900 \cos \theta+750 \sin \theta \\ N &=900 \cos 30^{\circ}+750 \sin 30^{\circ} \\ &=1154.4228 \mathrm{~N} \\ F_{\max }+900 \sin 30^{\circ} &=750 \cos 30^{\circ} \\ \mu N &=199.519 \\ \mu &=\frac{199.519}{1154.4228} \\ \mu &=0.1728 \end{aligned}
Question 4
A block of mass 10 kg rests on a horizontal floor. The acceleration due to gravity is 9.81 m/s^2. The coefficient of static friction between the floor and the block is 0.2. A horizontal force of 10 N is applied on the block as shown in the figure. The magnitude of force of friction (in N) on the block is___
A
5
B
10
C
12
D
18
GATE ME 2019 SET-1   Engineering Mechanics
Question 4 Explanation: 
Maximum friction force, f_{\max }=\mu \mathrm{N}=0.2 \times 10 \times 9.81=19.62 \mathrm{N}
Applied force, P=10 \mathrm{N} \lt \mathrm{f}_{\max }
\therefore Friction force = Applied force =10 \mathrm{N}
Question 5
A wardrobe (mass 100 kg, height 4 m, width 2 m, depth 1 m), symmetric about the Y-Y axis, stands on a rough level floor as shown in the figure. A force P is applied at mid-height on the wardrobe so as to tip it about point Q without slipping. What are the minimum values of the force (in newton) and the static coefficient of friction \mu between the floor and the wardrobe, respectively?
A
490.5 and 0.5
B
981 and 0.5
C
1000.5 and 0.15
D
1000.5 and 0.25
GATE ME 2014 SET-4   Engineering Mechanics
Question 5 Explanation: 
We can solve this question by moment or by force
triangle as this is condition of equilibrium (just to topple)
\begin{aligned} \sum M_{Q} &=0 \\ P \times 2-m g \times 1 &=0 \\ P &=\frac{m g}{2}=\frac{981}{2} \\ &=490.5 \mathrm{N} \end{aligned}
as P is horizontal force and it will be over come
by friction force
P=\left(f_{s}\right)_{\max } [ as this is at verge of motion ]
\begin{aligned} 490.5 &=\mu_{s} m g \\ \mu_{s} &=0.5 \end{aligned} OR

\begin{aligned} \tan \theta &=\frac{2}{1} \\ \theta &=63.4349^{\circ} \end{aligned}
Wardrobe is about to topple about pt Q so there
will be contact force from point Q on the wardrobe


\begin{aligned} \frac{W}{P} &=\tan \theta=2 \\ P &=\frac{W}{2}=\frac{981}{2}=490.5 \mathrm{N} \end{aligned}
Two components of R will balance W and P and
horizontal component (parallel to contacting
surface is called friction and perpendicular to
contacting surface is called normal reaction)
\begin{aligned} R \cos \theta &=\text { friction force } \\ &=P(\text { in magnitude }) \\ \left(f_{s}\right)_{\max } &=P=490.5 \\ \text{and}\qquad \left(f_{s}\right)_{\max } &=\mu_{s} m g\\ \mu_{s} m g &=490.5 \\ \mu_{s} &=0.5 \end{aligned}


There are 5 questions to complete.

9 thoughts on “Friction”

  3. Question no 10.
    Correct option is B = 0.8 N (Body is in equilibrium)
    How can friction be Zero , With Zero friction body will move on applying force

    Reply

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