GATE Mechanical Engineering Applied Mechanics and Design Syllabus

For a square-threaded screw jack, give the effort (torque) to raise a load $W$.

$T = W r \tan(\alpha + \phi)$, where $r$ is the mean thread radius, $\alpha = \tan^{-1}\!\big(\dfrac{p}{2\pi r}\big)$ is the helix angle, $\phi = \tan^{-1}\mu$ is the friction angle, $p$ the pitch (lead).

For a screw jack, give the torque to lower the load and the self-locking condition.

Lowering torque $T = W r \tan(\phi - \alpha)$. The screw is self-locking (load won't descend on its own) when $\phi \geq \alpha$, i.e. $\tan\alpha \leq \mu$.

State the efficiency of a square-threaded screw jack.

$\eta = \dfrac{\tan\alpha}{\tan(\alpha+\phi)}$. Maximum efficiency occurs at $\alpha = 45^\circ - \dfrac{\phi}{2}$, giving $\eta_{max} = \dfrac{1-\sin\phi}{1+\sin\phi}$.

Explain the mechanical principle of a wedge.

A wedge is a double inclined plane that converts a small applied force over a long distance into a large force (or displacement) over a short distance, used for lifting heavy loads or tightening; friction on all contact faces must be included in equilibrium.

In vehicle dynamics, what limits the maximum acceleration/braking on a level road?

Tire-road friction: the maximum traction/braking force at the driven/braked wheels cannot exceed $\mu N$ (friction coefficient times the normal load on those wheels). Hence $a_{max} \leq \mu g$ for a vehicle braking with all wheels.

How does weight transfer affect a braking vehicle?

During braking (deceleration $a$), inertia shifts load forward: normal force on front wheels increases and on rear wheels decreases by $\dfrac{m a h}{L}$ (h = CG height, L = wheelbase), increasing front grip and reducing rear grip.

What is the difference between a truss and a frame (in structural mechanics)?

A truss consists of two-force members joined at pin joints carrying only axial loads (tension/compression). A frame contains at least one multi-force member subjected to bending, shear, and axial loads, and is loaded at joints and along members.

State the determinacy condition for a planar (2D) pin-jointed truss.

$m + r = 2j$, where $m$ = number of members, $r$ = reaction components, $j$ = number of joints. If $m + r > 2j$ it is statically indeterminate; if $< 2j$ it is unstable (a mechanism).

Describe the method of joints for truss analysis.

Isolate each joint as a particle and apply $\sum F_x = 0$, $\sum F_y = 0$ (concurrent forces, no moment equation). Start at a joint with at most two unknown member forces and proceed through the truss.

Describe the method of sections for truss analysis.

Cut the truss with an imaginary section through the members of interest (at most three unknowns), then apply $\sum F_x=0,\ \sum F_y=0,\ \sum M=0$ to one portion to solve directly for specific member forces.

What is a zero-force member and how is it identified?

A member carrying no axial force. Identified by inspection: (1) at an unloaded two-member joint with non-collinear members, both are zero-force; (2) at an unloaded three-member joint where two are collinear, the third (non-collinear) member is zero-force.

State the principle of virtual work.

A system in equilibrium does zero total virtual work under any kinematically admissible virtual displacement: $\delta W = \sum \vec{F}_i \cdot \delta \vec{r}_i = 0$. For ideal (workless) constraints, only active forces contribute.

What is the main advantage of the virtual work method?

It allows solving for a single unknown force or equilibrium configuration without computing internal/constraint reactions, since workless constraint forces drop out — efficient for mechanisms and connected rigid-body systems.

State the relation among angular velocity, angular acceleration, and time for rigid-body rotation (constant $\alpha$).

$\omega = \omega_0 + \alpha t$, $\theta = \omega_0 t + \tfrac{1}{2}\alpha t^{2}$, $\omega^{2} = \omega_0^{2} + 2\alpha\theta$ — analogous to linear constant-acceleration kinematics.

Write the velocity relation between two points A and B on a rigid body in plane motion.

$\vec{v}_B = \vec{v}_A + \vec{\omega}\times \vec{r}_{B/A}$, where $\vec{\omega}$ is the body's angular velocity and $\vec{r}_{B/A}$ the position of B relative to A.

Write the acceleration relation between two points on a rigid body in plane motion.

$\vec{a}_B = \vec{a}_A + \vec{\alpha}\times\vec{r}_{B/A} + \vec{\omega}\times(\vec{\omega}\times\vec{r}_{B/A})$; the last term is the centripetal acceleration of magnitude $\omega^{2} r_{B/A}$ directed from B toward A.

What is the instantaneous centre of rotation (ICR) in plane motion?

The point in (or extended from) a rigid body that has zero velocity at a given instant; the body appears to rotate purely about it. It is found at the intersection of perpendiculars drawn to the velocity vectors of two points.

State the equations of motion for a rigid body in plane motion.

$\sum \vec{F} = m\,\vec{a}_G$ (translation of mass centre) and $\sum M_G = I_G\,\alpha$ (rotation about the mass centre), where $I_G$ is the centroidal mass moment of inertia.

State the parallel-axis theorem for mass moment of inertia.

$I = I_G + m d^{2}$, where $I_G$ is the moment of inertia about the centroidal axis, $m$ the mass, and $d$ the distance between the parallel axis and the centroidal axis.

State the impulse–momentum theorem (linear).

The linear impulse equals the change in linear momentum: $\int_{t_1}^{t_2}\vec{F}\,dt = m\vec{v}_2 - m\vec{v}_1$. With no external force, linear momentum is conserved.

State the angular impulse–momentum theorem.

$\int_{t_1}^{t_2} \vec{M}_O\,dt = \vec{H}_{O2} - \vec{H}_{O1}$, where $\vec{H}_O$ is angular momentum about O. With no external moment, angular momentum is conserved (e.g. spinning skater).

Define the coefficient of restitution and give the velocity relation in an impact.

$e = \dfrac{\text{relative velocity of separation}}{\text{relative velocity of approach}} = \dfrac{v_2' - v_1'}{v_1 - v_2}$ (along the line of impact). $e=1$ perfectly elastic, $e=0$ perfectly plastic, $0<e<1$ real.

Write the kinetic energy of a rigid body in general plane motion.

$T = \tfrac{1}{2} m v_G^{2} + \tfrac{1}{2} I_G \omega^{2}$ — sum of translational KE of the mass centre and rotational KE about the centroidal axis.

State the work–energy principle for a rigid body.

The total work done by all external forces and couples equals the change in kinetic energy: $U_{1\to2} = T_2 - T_1$. For conservative systems, $T_1 + V_1 = T_2 + V_2$.

State the Lagrange equation of motion for a holonomic system.

$\dfrac{d}{dt}\!\left(\dfrac{\partial L}{\partial \dot{q}_i}\right) - \dfrac{\partial L}{\partial q_i} = Q_i$, where $L = T - V$ is the Lagrangian, $q_i$ generalized coordinates, and $Q_i$ generalized non-conservative forces (zero for conservative systems).

In Lagrangian mechanics, what is a generalized coordinate and how many are needed?

An independent variable describing the configuration of a system. The number required equals the number of degrees of freedom (total coordinates minus the number of independent holonomic constraints).

Define normal stress and shear stress.

Normal stress $\sigma = \dfrac{P}{A}$ acts perpendicular to the cross-section (tension or compression); shear stress $\tau = \dfrac{V}{A}$ acts tangential (parallel) to the section. SI unit: pascal (Pa).

Define normal strain and shear strain.

Normal strain $\varepsilon = \dfrac{\Delta L}{L}$ is fractional change in length (dimensionless). Shear strain $\gamma$ is the change in the right angle between two originally perpendicular line elements, in radians.

State Hooke's law and define the elastic constants $E$, $G$, and $K$.

In the linear-elastic range stress is proportional to strain. $E$ (Young's modulus): $\sigma = E\varepsilon$; $G$ (shear/rigidity modulus): $\tau = G\gamma$; $K$ (bulk modulus): $\sigma_{vol} = -K\,\dfrac{\Delta V}{V}$.

Define Poisson's ratio and give its typical range.

$\nu = -\dfrac{\text{lateral strain}}{\text{longitudinal strain}}$. For isotropic materials thermodynamic limits are $-1 < \nu < 0.5$; most metals have $\nu \approx 0.25$–$0.35$. Incompressible materials have $\nu = 0.5$.

Give the relations among $E$, $G$, $K$, and $\nu$ for an isotropic material.

$E = 2G(1+\nu)$, $E = 3K(1-2\nu)$, and combining, $E = \dfrac{9KG}{3K+G}$. Only two of the four constants are independent.

For plane stress, give the principal stresses in terms of $\sigma_x,\sigma_y,\tau_{xy}$.

$\sigma_{1,2} = \dfrac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}$, and maximum in-plane shear $\tau_{max} = \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^{2} + \tau_{xy}^{2}} = \dfrac{\sigma_1 - \sigma_2}{2}$.

What does Mohr's circle represent, and what are its centre and radius for plane stress?

A graphical plot of normal stress (x-axis) vs shear stress (y-axis) for all plane orientations. Centre $= \left(\dfrac{\sigma_x+\sigma_y}{2},\,0\right)$; radius $R = \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^{2}+\tau_{xy}^{2}} = \tau_{max}$.

In Mohr's circle, how do angles on the circle relate to physical plane orientations?

Angles on Mohr's circle are twice the physical angles: a rotation of $\theta$ in the material corresponds to $2\theta$ on the circle, and rotation sense is preserved. Principal planes (where $\tau=0$) are $90^\circ$ apart on the circle ($45^\circ$ physically from max-shear planes).