Question

A uniforin wooden stick of mass \(1.6 \mathrm{~kg}\) and length \(i\) rests in an inclined manner on a smooth, vertical wall of height \(h(

Short answer

The correct option is (C) where \frac{h}{l}=\frac{3 \times \frac{\text{sin}(30)}{\text{cos}(30)}}{16}, f=\frac{8 \times \text{sqrt}(3)}{3} \text{ N}.

Step-by-step solution

Understand the Situation and Draw a Free Body Diagram

First, imagine the stick resting against the wall, making a 30-degree angle with it. Draw a diagram of the stick with forces acting on it: the weight (W) acting downwards at its center of mass, the normal force by the wall (Fw) perpendicular to the stick, the normal force by the floor (Ff), and the friction force (f) at the floor.

Apply Conditions for Equilibrium

Since the stick is at rest, it is in equilibrium. This means the sum of the forces in all directions must be zero. Horizontally, the friction force (f) balances the horizontal component of Fw. Vertically, the normal force by the floor (Ff) balances the weight (W) and the vertical component of Fw.

Find Horizontal Components of the Forces

The force of the wall Fw acts at an angle of 30 degrees to the vertical. Thus, its horizontal component which is balanced by friction (f) is Fw*sin(30). Its vertical component is Fw*cos(30).

Analyse Vertical Forces and Set Up Equations

In the vertical direction, Ff (reaction from the floor) must balance both the weight W of the stick and the vertical component of Fw: Ff = W + Fw*cos(30).

Solve for Fw and f Using Equilibrium Conditions

The conditions of equilibrium yield Fw = f. Also, W of the stick is 1.6 kg * 10 m/s^2 (gravitational acceleration) = 16 N. Therefore, Fw*cos(30) = 16 N - Ff = 16 N - Fw. Solving for Fw gives Fw = f = 16 / (1+cos(30)) N.

Calculate the Frictional Force f

Having Fw, calculate f using the expression found for Fw: f = 16 / (1 + cos(30)) N. Calculate cos(30) which is sqrt(3)/2 and substitute back into the equation to find the exact value of f.

Find the Ratio h/l

To find the ratio h/l, realize that the vertical component of the length of the stick above the floor is h and the full length of the stick is l. Since the stick makes an angle of 30 degrees with the wall, h/l = sin(30). Calculate sin(30) to get the ratio.

Key concepts

Equilibrium in Mechanics

Equilibrium in mechanics is a fundamental concept that refers to the state of a system where all forces and moments are balanced, leading to no acceleration. In other words, for an object to be in equilibrium, the net force and net torque acting on it must be zero. This crucial principle allows us to examine situations where objects remain at rest (static equilibrium) or move with a constant velocity (dynamic equilibrium).

To determine if an object is in equilibrium, we look at the forces acting on the object and apply Newton's first law of motion. According to this law, an object at rest will stay at rest, and an object in motion will stay in motion with a constant speed and direction unless acted on by an unbalanced external force. When an object is in a state of equilibrium, the various forces leading to its rest or constant motion are said to be balanced.

For the wooden stick in the problem, equilibrium means it does not slide down the wall or fall over; the forces acting horizontally and vertically must cancel out. To solve for equilibrium in mechanics, it’s crucial to identify all the forces acting on the body, which often involves drawing a free body diagram.

Free Body Diagram

A free body diagram (FBD) is a visual representation used to depict all the external forces acting upon an object. In physics problem solving, the FBD is an essential step as it helps isolate the object of interest and simplifies complex interactions with its environment into manageable components. When creating a free body diagram, each force is represented by an arrow pointing in the direction that force is applied, and the length of the arrow visually represents the magnitude of the force.

In the case of the wooden stick resting against the wall, the FBD serves as a tool to understand how the stick maintains its equilibrium. When analyzing the stick, we would draw the weight of the stick acting downwards at its center of mass, the normal force by the wall perpendicular to the stick, and the normal and frictional forces at the point of contact with the floor. By breaking down the forces into their individual components, the FBD allows for a clearer view to apply the conditions for equilibrium. In classroom settings or homework exercises, the FBD can dramatically improve the comprehension and problem-solving process.

Frictional Force

Frictional force is the force exerted by a surface as an object moves across it or makes an effort to move across it. It is always directed opposite to the object's motion or intended motion. Frictional forces arise due to the irregularities found on the two surfaces in contact and the intermolecular forces between the particles of the surfaces.

In many physics problems, including the one at hand, the frictional force plays a crucial role in preventing sliding motion and is essential to maintaining equilibrium. The amount of frictional force that develops between surfaces is typically proportional to the normal force acting on the object from the surface, and it is characterized by the coefficient of friction specific to the pair of materials in contact.

In the example problem of the wooden stick, the frictional force at the base resists the horizontal component of the reaction force from the wall, compensating for the tendency of the stick to slide. Without sufficient friction, the stick would not be able to maintain its tilted position and would fall over. Computing frictional force, as seen in the step by step solution, involves understanding both the normal force and the component of any other force acting along the surface.