91Ó°ÊÓ

One end of a uniform meter stick is placed against a vertical wall (Fig. P11. 70). The other end is held by a lightweight cord that makes an angle \(\theta\) with the stick. The coefficient of static friction between the end of the meter stick and the wall is 0.40 . (a) What is the maximum value the angle \(\theta\) can have if the stick is to remain in equilibrium? (b) Let the angle \(\theta\) be \(15^{\circ} .\) A block of the same weight as the meter stick is suspended from the stick, as shown, at a distance \(x\) from the wall. What is the minimum valueof \(x\) for which the stick will remain in equilibrium? (c) When \(\theta=15^{\circ},\) how large must the coefficient of static friction be so that the block can be attached 10 \(\mathrm{cm}\) from the left end of the stick without causing it to slip?

Step by step solution

01

Identify forces acting on the stick

Consider the forces acting on the stick: the tension \( T \) from the cord, the frictional force \( f \) from the wall, the normal force \( N \) from the wall, and the gravitational force \( mg \) (where \( m \) is mass and \( g \) is gravity) acting at the stick's center.

02

Set up the equations for equilibrium

For the stick to be in equilibrium, the sum of forces and the sum of torques must be zero. 1. Horizontal force balance: \( T \cos\theta = f \)2. Vertical force balance: \( T \sin\theta + N = mg \)

03

Consider friction related to normal force

Use the coefficient of static friction \( \mu = 0.4 \) to express the maximum static friction as \( f = \mu N \). Substitute into the force balance equations.

04

Set up the torque balance equation

Take torques about the point where the stick contacts the wall to eliminate \( N \) and \( f \). Use \( mg \times (l/2) \) and solve for \( T \), where \( l \) is the length of the stick.

05

Solve Part (a) for maximum \( \theta \)

Substitute known values into the equations derived and isolate \( \theta \). Use trigonometric identities to find \( \theta_{max} \).

06

Assume conditions for block addition (Part b)

For part (b), with the block's weight being added, re-derive equilibrium conditions including the additional force \( mg \) at location \( x \) from the pivot.

07

Solve Part (b) for minimum \( x \)

Solve the derived torque and equilibrium conditions from step 6 to find the minimum \( x \) in terms of a specific angle \( \theta=15^{\circ} \).

08

Consider conditions for modified friction (Part c)

Adjust the friction coefficient condition \( \mu \) for the given position \( x = 10 \) cm and \( \theta = 15^{\circ} \), maintaining balance with normalized forces.

09

Solve Part (c) for \( \mu \)

Substitute the angle and position from previous steps into new balance equations, solving for the minimum static coefficient necessary to prevent slipping.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statics

Statics is the study of forces and their effects on a system in equilibrium. An object is in equilibrium when it remains at rest or moves with constant velocity. For the meter stick in the exercise, being in equilibrium means that the stick does not rotate or translate. To analyze the problem:

• Identify all forces acting on the stick. These include tension from the cord, friction at the wall, normal force at the wall, and gravity acting through the stick's center of mass.
• Ensure the sum of all horizontal and vertical forces equals zero. This prevents any linear movement.
• Ensure the sum of all torques about any point is zero. This prevents the stick from rotating.

By carefully analyzing these conditions, one can determine how to keep the stick from slipping or tipping, achieving a perfect state of equilibrium.

Torque

Torque is a measure of the rotational force applied to an object. It's crucial in understanding equilibrium, as forces not only push or pull objects but can also cause them to rotate. Torque depends on the force's magnitude, the distance from the pivot point, and the angle at which it is applied.To calculate torque (\[ \tau \]), use the formula:- \[ \tau = r \cdot F \cdot \sin(\theta) \] where:

• \( r \) is the distance from the pivot point.
• \( F \) is the force applied.
• \( \theta \) is the angle between the force direction and the line through the pivot point.

In the exercise, one must calculate torques about a point (often where the stick contacts the wall) to ensure equilibrium. By balancing torques, you can solve for unknown forces or angles, ensuring no rotation occurs and maintaining equilibrium.

Friction

Friction is the force that opposes the relative motion or tendency of such motion of two surfaces in contact. In the meter stick exercise, static friction plays a critical role in keeping the stick from slipping down the wall. The static friction depends on the coefficient of static friction (\( \mu \)) and the normal force (\( N \)) exerted by the wall:- \[ f = \mu \cdot N \]This force acts horizontally at the contact point between the stick and the wall, countering the component of the weight wanting to slide the stick down. Static friction must not exceed this limit, or the stick will slip. To ensure equilibrium:

• Confirm the calculated friction does not surpass \( \mu N \).
• Use it to find the maximum angle \( \theta \) for which the stick can be balanced.

Friction analysis helps determine critical conditions under which the stick remains static.

Force Analysis

Force analysis involves breaking down all forces acting on an object and studying their contributions to equilibrium. Each force has both magnitude and direction that must be considered when analyzing its effect.To perform a force analysis:

• Break forces into their components, typically horizontal and vertical.
• Set up equations that sum horizontal forces to zero: \[ T \cdot \cos(\theta) = f \]
• Set up equations that sum vertical forces to zero: \[ T \cdot \sin(\theta) + N = mg \]

This study allows you to re-derive conditions when additional forces, like an extra block, are introduced. Adjust your calculations to consider these changes, keeping the stick balanced even as other factors vary. Always remember, the goal is to stabilize all these forces to maintain a state of equilibrium.