Question 1

Pendulum terminology in engineering mechanics A rigid body suspended vertically from a fixed point and oscillating with a small amplitude under gravity is termed what type of pendulum?

Accepted Answer

Introduction / Context:In engineering mechanics and vibrations, the term used for a rigid body swinging about a fixed horizontal axis under gravity is important for selecting correct formulas for time period and center of oscillation. Many learners confuse “simple pendulum” with “compound” (also called “physical”) pendulum.Given Data / Assumptions:The body is rigid (not a point mass and not a massless string).It is suspended to swing about a fixed horizontal axis under gravity.Amplitude is small so the small-angle approximation applies.Concept / Approach:A simple pendulum is an ideal model: a point mass suspended by a weightless inextensible string. A compound pendulum (also called a physical pendulum) is any rigid body oscillating about a fixed horizontal axis under gravity. Its dynamics depend on the body’s mass moment of inertia about the suspension axis and the distance of its center of gravity from that axis.Step-by-Step Solution:Identify the physical system: a rigid body, not a point mass + string.Identify motion: small angular oscillations about a fixed axis under gravity.Match with definition: that is the definition of a compound (physical) pendulum.Verification / Alternative check:The time period for a compound pendulum is T = 2 * pi * sqrt(I_o / (m * g * h)), where I_o is the mass moment of inertia about the suspension axis and h is the distance from the suspension axis to the center of gravity. This formula explicitly requires a rigid body, confirming the classification.Why Other Options Are Wrong:Simple pendulum assumes a point mass and string; second's pendulum is a special simple pendulum with T = 2 s; “none” is incorrect because the standard term exists.Common Pitfalls:Interchanging “simple” and “compound”; overlooking that “physical pendulum” is synonymous with “compound pendulum”.Final Answer:Compound pendulum

Question 2

Friction on a ladder: rough ground and smooth wall A ladder rests on rough ground and leans against a perfectly smooth vertical wall. At the upper end (at the wall), the force of friction acts in which manner?

Accepted Answer

Introduction / Context:Ladder friction problems test understanding of contact forces at rough and smooth interfaces. When one contact is smooth, it cannot transmit tangential force (i.e., no friction), which simplifies equilibrium equations and determines where friction must act.Given Data / Assumptions:Ground is rough (friction possible at the foot).Wall is perfectly smooth (no friction at the top contact).Rigid ladder; static equilibrium; no impending motion unless specified.Concept / Approach:A smooth surface can only exert a normal reaction, perpendicular to itself. Therefore, at the smooth vertical wall, the contact force is purely horizontal (normal to the wall) and has no vertical component and no friction. The rough ground must provide the necessary frictional force to prevent slipping in the vertical direction and maintain equilibrium of the ladder’s weight and components of forces.Step-by-Step Solution:At wall: reaction R_w is horizontal; friction F_w = 0 because the wall is smooth.At ground: normal reaction R_g is vertical; friction F_g is horizontal to balance the horizontal reaction at the wall and to prevent slipping.Sum of forces and moments confirm that only the ground contact supplies frictional resistance.Verification / Alternative check:Draw a free-body diagram. If any nonzero vertical friction existed at the wall, it would be a tangential component; smoothness forbids it. Hence friction at the top is zero.Why Other Options Are Wrong:(a) and (b) imply vertical friction at the wall, impossible for a smooth surface. (c) describes the normal reaction, not friction. (e) is not a valid wall-contact friction direction here.Common Pitfalls:Assuming some friction at the wall; mixing the roles of ground and wall contacts; forgetting to align reactions perpendicular to their surfaces.Final Answer:Zero at its upper end

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