Engineering Mechanics
According to the Parallelogram Law, the magnitude of the resultant (R) of two forces (P and Q) with an angle θ between them is:
- \( R = P + Q \)
- \( R = \sqrt{P^2 + Q^2 + 2PQ\cos\theta} \)
- \( R = \sqrt{P^2 + Q^2} \)
- \( R = P + Q\cos\theta \)
Explanation:
- The correct formula is derived from the law of cosines. The other options are incorrect: \( R = P + Q \) assumes collinear, same-direction forces; \( R = \sqrt{P^2 + Q^2} \) is only true for θ = 90°; and \( R = P + Q\cos\theta \) is not a standard resultant formula.
Lami’s theorem, used for three concurrent forces in equilibrium (P, Q, R), is expressed as:
- \( \frac{P}{\cos\alpha} = \frac{Q}{\cos\beta} = \frac{R}{\cos\gamma} \)
- \( P\sin\alpha = Q\sin\beta = R\sin\gamma \)
- \( \frac{P}{\sin\alpha} = \frac{Q}{\sin\beta} = \frac{R}{\sin\gamma} \)
- \( P + Q + R = 0 \)
Explanation:
- Lami’s theorem states that for three concurrent and coplanar forces in equilibrium, each force is proportional to the sine of the angle between the other two. Here, α, β, and γ are the angles opposite to forces P, Q, and R, respectively.
The moment (M) of a couple, defined by two forces (F) separated by a perpendicular distance (d), is given by:
- \( M = F \cdot d \)
- \( M = F \cdot d \)
- \( M = \frac{1}{2}F \cdot d \)
- \( M = F + d \)
Explanation:
- A couple consists of two equal, opposite, and parallel forces. Its moment is the product of one force and the perpendicular distance between their lines of action. This moment is a pure rotational effect.
The relationship between the coefficient of static friction (μ) and the limiting angle of friction (φ) is:
- \( \mu = \sin\phi \)
- \( \mu = \cos\phi \)
- \( \mu = \tan\phi \)
- \( \mu = \cot\phi \)
Explanation:
- From the geometry of the friction triangle, the coefficient of friction (μ) is equal to the ratio of the frictional force (F) to the normal reaction (N). This ratio is also the tangent of the angle of friction: \( \mu = F/N = \tan\phi \).
The general conditions for the equilibrium of a rigid body subjected to coplanar, non-concurrent forces are:
- \( \sum F_x = 0 \) and \( \sum F_y = 0 \)
- \( \sum F_x = 0 \) and \( \sum M = 0 \)
- \( \sum F_x = 0 \), \( \sum F_y = 0 \), and \( \sum M = 0 \)
- \( \sum M = 0 \) about two different points
Explanation:
- For complete equilibrium, the body must have no linear acceleration (ensured by \( \sum F_x = 0 \) and \( \sum F_y = 0 \)) and no angular acceleration (ensured by \( \sum M = 0 \) about any point).
The moment of inertia (I) of a particle of mass (m) about an axis is defined as:
- \( I = m \cdot r \)
- \( I = m \cdot r^2 \)
- \( I = \frac{1}{2}m \cdot r^2 \)
- \( I = m \cdot g \cdot r \)
Explanation:
- The moment of inertia is the sum of the product of the mass of each particle and the square of its perpendicular distance from the axis of rotation. For a single particle, it is \( I = m r^2 \).
D’Alembert’s principle converts a problem in dynamics into an equivalent problem in statics by:
- Applying the principle of conservation of energy
- Resolving forces into components
- Introducing an inertial force (-m·a) to the system
- Applying the principle of virtual work
Explanation:
- D’Alembert’s principle states that the sum of the differences between the applied forces and the inertial forces (ma) for a system is zero. Adding the fictitious inertial force (-ma) allows the equations of motion to be written as equations of static equilibrium: \( \sum F – ma = 0 \).
The equation of the path (trajectory) of a projectile is:
- Linear
- Circular
- Parabolic
- Elliptical
Explanation:
- The trajectory is derived from the horizontal motion, \( x = (u\cos\theta)t \), and vertical motion, \( y = (u\sin\theta)t – \frac{1}{2}gt^2 \). Eliminating time (t) gives an equation of the form \( y = x\tan\theta – \frac{gx^2}{2u^2\cos^2\theta} \), which is the equation of a parabola.