# GATE Questions & Answers of Theory of Machines Mechanical Engineering

#### Theory of Machines 72 Question(s) | Weightage 10 (Marks)

A four bar mechanism is made up of links of length 100, 200, 300 and 350 mm. If the 350 mm link is fixed, the number of links that can rotate fully is __________.

An epicyclic gear train is shown in the figure below. The number of teeth on the gears A, B and D are 20, 30 and 20, respectively. Gear C has 80 teeth on the inner surface and 100 teeth on the outer surface. If the carrier arm AB is fixed and the sun gear A rotates at 300 rpm in the clockwise direction, then the rpm of D in the clockwise direction is

A slider crank mechanism is shown in the figure. At some instant, the crank angle is $45^\circ$ and a force of 40 N is acting towards the left on the slider. The length of the crank is 30 mm and the connecting rod is 70 mm. Ignoring the effect of gravity, friction and inertial forces, the magnitude of the crankshaft torque (in Nm) needed to keep the mechanism in equilibrium is _________ (correct to two decimal places).

A frictionless gear train is shown in the figure. The leftmost 12-teeth gear is given a torque of 100 N-m. The output torque from the 60-teeth gear on the right in N-m is

In a cam-follower, the follower rises by $h$ as the cam rotates by $\delta$ (radians) at constant angular velocity $\omega$ (radians/s). The follower is uniformly accelerating during the first half of the rise period and it is uniformly decelerating in the latter half of the rise period. Assuming that the magnitudes of the acceleration and deceleration are same, the maximum velocity of the follower is

In an epicyclic gear train, shown in the figure, the outer ring gear is fixed, while the sun gear rotates counterclockwise at 100 rpm. Let the number of teeth on the sun, planet and outer gears to be 50, 25, and 100, respectively. The ratio of magnitudes of angular velocity of the planet gear to the angular velocity of the carrier arm is ____________.

For an inline slider-crank mechanism, the length of the crank and connecting rod are 3 m and 4 m, respectively. At the instant when the connecting rod is perpendicular to the crank, if the velocity of the slider is 1 m/s, the magnitude of angular velocity (upto 3 decimal points accuracy) of the crank is _________radian/s.

In a slider-crank mechanism, the length of the crank and the connecting rod are 100 mm and 160 mm, respectively. The crank is rotating with an angular velocity 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__________

Block 2 slides outward on link 1 at a uniform velocity of 6 m/s as shown in figure. Link 1 is rotating at a constant angular velocity of 20 radian/s  counterclockwise. The magnitude of the total accelecration (in m/s2) of point P of the block with respect to fixed point O is___________

A gear train shown in the figure consist of gears P, Q, R and S. Gear Q and gear R are mounted on the same shaft. All the gears are mounted on parallel shaft and the number of teeth of P, Q, R and S are 24, 45, 30 and 80, respectively. Gear P is rotating at 400 rpm. The speed (in rpm) of the gear S is_________

A car is moving on a curved horizontal road of radius 100 m with a speed of 20 m/s. The rotating masses of the engine have an angular speed of 100 rad/s in clockwise direction when viewed from the front of the car. The combined moment of inertia of the rotating masses is 10 kg-$\mathrm m^2$. The magnitude of the gyroscopic moment (in N-m) is __________

A slider crank mechanism with crank radius 200 mm and connecting rod length 800 mm is shown. The crank is rotating at 600 rpm in the counterclockwise direction. In the configuration shown, the crank makes an angle of $90^\circ$ with the sliding direction of the slider, and a force of 5 kN is acting on the slider. Neglecting the inertia forces, the turning moment on the crank (in kN-m) is __________

In the gear train shown, gear 3 is carried on arm 5. Gear 3 meshes with gear 2 and gear 4. The number of teeth on gear 2, 3, and 4 are 60, 20, and 100, respectively. If gear 2 is fixed and gear 4 rotates with an angular velocity of 100 rpm in the counterclockwise direction, the angular speed of arm 5 (in rpm) is

The forces $F_1$ and $F_2$ in a brake band and the direction of rotation of the drum are as shown in the figure. The coefficient of friction is 0.25. The angle of wrap is $3\mathrm\pi/2$ radians. It is given that $R=1$ m and $F_2=1\;N$. The torque (in N-m) exerted on the drum is _________

For the brake shown in the figure, which one of the following is TRUE?

The rod AB, of length 1 m, shown in the figure is connected to two sliders at each end through pins. The sliders can slide along QP and QR. If the velocity $V_A$ of the slider at A is 2 m/s, the velocity of the midpoint of the rod at this instant is ___________ m/s.

The number of degrees of freedom in a planar mechanism having $n$ links and $j$ simple hinge joints is

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.

Consider a slider crank mechanism with nonzero masses and inertia. A constant torque $\tau$ is applied on the crank as shown in the figure. Which of the following plots best resembles variation of crank angle , $\theta$ versus time

The number of degrees of freedom of the planetary gear train shown in the figure is

In a certain slider-crank mechanism, lengths of crank and connecting rod are equal. If the crank rotates with a uniform angular speed of 14 rad/s and the crank length is 300 mm, the maximum acceleration of the slider (in m/s2) is _______.

For the same material and the mass, which of the following configurations of flywheel will have maximum mass moment of inertia about the axis of rotation OO′ passing through the center of gravity.

A gear train is made up of five spur gears as shown in the figure. Gear 2 is driver and gear 6 is driven member. N2, N3, N4, N5 and N6 represent number of teeth on gears 2,3,4,5, and 6 respectively, The gear(s) which act(s) as idler(s) is/ are

In the figure, link 2 rotates with constant angular velocity $\omega$2. A slider link 3 moves outwards with a constant relative velocity VQ/P, where Q is a point on slider 3 and P is a point on link 2. The magnitude and direction of Coriolis component of acceleration is given by

The number of degrees of freedom of the linkage shown in the figure is

The torque (in N-m) exerted on the crank shaft of a two stroke engine can be described as T = 10000 + 1000 sin 2θ – 1200 cos 2θ, where θ is the crank angle as measured from inner dead center position. Assuming the resisting torque to be constant, the power (in kW) developed by the engine at 100 rpm is __________.

For the given statements:

 I. Mating spur gear teeth is an example of higher pair II. A revolute joint is an example of lower pair

Indicate the correct answer.

A rigid link PQ is 2 m long and oriented at 20° to the horizontal as shown in the figure.The magnitude and direction of velocity VQ,and the direction of velocity VP are given.The magnitude of VP (in m/s) at this instant is

A pair of spur gears with module 5 mm and a center distance of 450 mm is used for a speed reduction of 5:1. The number of teeth on pinion is _______

A slider crank mechanism has slider mass of 10 kg, stroke of 0.2 m and rotates with a uniform angular velocity of 10 rad/s. The primary inertia forces of the slider are partially balanced by a revolving mass of 6 kg at the crank, placed at a distance equal to crank radius. Neglect the mass of connecting rod and crank. When the crank angle (with respect to slider axis) is 30°, the unbalanced force (in newton) normal to the slider axis is _______

An offset slider-crank mechanism is shown in the figure at an instant. Conventionally, the Quick Return Ratio (QRR) is considered to be greater than one. The value of QRR is _______

A 4-bar mechanism with all revolute pairs has link lengths lf = 20 mm, lin = 40 mm, lco = 50 mm and lout = 60 mm. The suffixes 'f', 'in', 'co' and 'out' denote the fixed link, the input link, the coupler and output link respectively. Which one of the following statements is true about the input and output links?

Consider a flywheel whose mass M is distributed almost equally between a heavy, ring-like rim of radius R and a concentric disk-like feature of radius R/2. Other parts of the flywheel, such as spokes, etc, have negligible mass. The best approximation for α, if the moment of inertia of the flywheel about its axis of rotation is expressed as αMR2, is _______

Maximum fluctuation of kinetic energy in an engine has been calculated to be 2600 J. Assuming that the engine runs at an average speed of 200 rpm, the polar mass moment of inertia (in kg.m2) of a flywheel to keep the speed fluctuation within ±0.5% of the average speed is _______

A rigid link PQ of length 2 m rotates about the pinned end Q with a constant angular acceleration of 12 rad/s2. When the angular velocity of the link is 4 rad/s, the magnitude of the resultant acceleration (in m/s2) of the end P is _______

A spur pinion of pitch diameter 50 mm rotates at 200 rad/s and transmits 3 kW power. The pressure angle of the tooth of the pinion is 20°. Assuming that only one pair of the teeth is in contact, the total force (in newton) exerted by a tooth of the pinion on the tooth on a mating gear is _______

Gear 2 rotates at 1200 rpm in counter clockwise direction and engages with Gear 3. Gear 3 and Gear 4 are mounted on the same shaft. Gear 5 engages with Gear 4. The numbers of teeth on Gears 2, 3, 4 and 5 are 20, 40, 15 and 30, respectively. The angular speed of Gear 5 is

Consider a rotating disk cam and a translating roller follower with zero offset. Which one of the following pitch curves, parameterized by t, lying in the interval 0 to 2π, is associated with the maximum translation of the follower during one full rotation of the cam rotating about the center at (x, y) = (0, 0) ?

A slider-crank mechanism with crank radius 60 mm and connecting rod length 240 mm is shown in figure. The crank is rotating with a uniform angular speed of 10 rad/s, counter clockwise. For the given configuration, the speed (in m/s) of the slider is _______

Which one of the following is used to convert a rotational motion into a translational motion?

It is desired to avoid interference in a pair of spur gears having a 20° pressure angle. With increase in pinion to gear speed ratio, the minimum number of teeth on the pinion

A uniform slender rod (8 m length and 3 kg mass) rotates in a vertical plane about a horizontal axis 1 m from its end as shown in the figure. The magnitude of the angular acceleration (in rad/s2) of the rod at the position shown is _______

Torque and angular speed data over one cycle for a shaft carrying a flywheel are shown in the figures. The moment of inertia (in kg.m2) of the flywheel is _______

A link OB is rotating with a constant angular velocity of 2 rad/s in counter clockwise direction and a block is sliding radially outward on it with an uniform velocity of 0.75 m/s with respect to the rod, as shown in the figure below. If OA = 1 m, the magnitude of the absolute acceleration of the block at location A in m/s2 is

A planar closed kinematic chain is formed with rigid links PQ = 2.0 m, QR = 3.0 m, RS = 2.5 m and SP = 2.7 m with all revolute joints. The link to be fixed to obtain a double rocker (rockerrocker) mechanism is

A flywheel connected to a punching machine has to supply energy of 400 Nm while running at a mean angular speed of 20 rad/s. If the total fluctuation of speed is not to exceed ±2%, the mass moment of inertia of the flywheel in kg-m2 is

A compound gear train with gears P, Q, R and S has number of teeth 20, 40, 15 and 20, respectively. Gears Q and R are mounted on the same shaft as shown in the figure below. The diameter of the gear Q is twice that of the gear R. If the module of the gear R is 2 mm, the center distance in mm between gears P and S is

The following are the data for two crossed helical gears used for speed reduction:
Gear I : Pitch circle diameter in the plane of rotation 80 mm and helix angle 30º
Gear II : Pitch circle diameter in the plane of rotation 120 mm and helix angle 22.5º
If the input speed is 1440 rpm, the output speed in rpm is

A solid disc of radius $r$ rolls without slipping on a horizontal floor with angular velocity $\alpha$ and angular acceleration $\alpha$. The magnitude of the acceleration of the point of contact on the disc is

In the mechanism given below, if the angular velocity of the eccentric circular disc is 1 rad/s, the angular velocity (rad/s) of the follower link for the instant shown in the figure is

A circular solid disc of uniform thickness 20 mm, radius 200 mm and mass 20 kg, is used as a flywheel. If it rotates at 600 rpm, the kinetic energy of the flywheel, in Joules is

A double – parallelogram mechanism is shown in the figure. Note that PQ is a single link. The mobility of the mechanism is

For the four-bar linkage shown in the figure, the angular velocity of link AB is 1 rad/s. The length of link CD is 1.5 times the length of link AB. In the configuration shown, the angular velocity of link CD in rad/s is

Mobility of a statically indeterminate structure is

There are two points P and Q on a planar rigid body. The relative velocity between the two points

Which of the following statements is INCORRECT?

Tooth interference in an external involute spur gear pair can be reduced by

For the epicyclic gear arrangement shown in the figure, ω2 =100rad / s clockwise (CW) and ωarm =80rad / s counter clockwise (CCW). The angular velocity ω5 (in rad/s)

For the configuration shown, the angular velocity of link AB is 10 rad/s counterclockwise. The magnitude of the relative sliding velocity (in ms-1) of slider B with respect to rigid link CD is

A simple quick return mechanism is shown in the figure. The forward to return ratio of the quick return mechanism is 2:1. If the radius of the crank O1P is 125 mm, then the distance 'd' (in mm) between the crank centre to lever pivot centre point should be

An epicyclic gear train is shown schematically in the adjacent figure.

The sun gear 2 on the input shaft is a 20 teeth external gear. The planet gear 3 is a 40 teeth external gear. The ring gear 5 is a 100 teeth internal gear. The ring gear 5 is fixed and the gear 2 is rotating at 60 rpm ccw (ccw=counterclockwise and cw=clockwise).

The arm 4 attached to the output shaft will rotate at

Match the approaches given below to perform stated kinematics / dynamics analysis of machine.

 Analysis Approach P. Continuous relative rotation 1. D’ Alembert’s principle Q. Velocity and acceleration 2. Grubler’s criterion R. Mobility 3. Grashoff’s law S. Dynamic-static analysis 4. Kennedy’s theorem

A planner mechanism has 8 links and 10 rotary joints. The number of degrees of freedom of the mechanism, using Gruebler’s criterion, is

Match the type of gears with their most appropriate description.

 Type of gear Description P Helical 1 Axes non parallel and non intersection Q Spiral Bevel 2 Axes parallel and teeth are inclined to the axis R Hypoid 3 Axes are parallel and teeth are parallel to the axis S Rack and pinion 4 Axes are perpendicular and intersecting, and teeth are inclined to the axis 5 Axes are perpendicular and used for large speed reduction 6 Axes parallel and one of the gears has infinite radius

One tooth of a gear having 4 module and 32 teeth is shown in the figure. Assume that the gear tooth and the corresponding tooth space make equal intercepts on the pitch circumference. The dimensions ‘a’ and ‘b’, respectively, are closed to

The input link O2P of a four bar linkage is rotated at 2 rad/s counter clockwise direction as shown below. The angular velocity of the coupler PQ in rad/s, at an instant when $\angle$O4O2P = 180°, is

The speed of an engine varies from 210 rad/s to 190 rad/s. During a cycle the change in kinetic energy is found to be 400 Nm. The inertia of the flywheel in kgm2 is

A gear set has a pinion with 20 teeth and a gear with 40 teeth. The pinion runs at 30 rev/s and transmits a power of 20 kW. The teeth are on the 20° full – depth system and have a module of 5 mm. The length of the line of action is 19 mm.

The center distance for the above gear set in mm is

A gear set has a pinion with 20 teeth and a gear with 40 teeth. The pinion runs at 30 rev/s and transmits a power of 20 kW. The teeth are on the 20° full – depth system and have a module of 5 mm. The length of the line of action is 19 mm.

The contact ratio of the contacting tooth is

A gear set has a pinion with 20 teeth and a gear with 40 teeth. The pinion runs at 30 rev/s and transmits a power of 20 kW. The teeth are on the 20° full – depth system and have a module of 5 mm. The length of the line of action is 19 mm.

The resultant force on the contacting gear tooth in N is

A quick return mechanism is shown below. The crank OS is driven at 2 rev / s in counter – clockwise direction.

If the quick return ratio is 1:2, then the length of the crank in mm is