Mechanical Properties Of Fluid

Step 3: Maximum tension condition

The maximum tension the rope can sustain is . Using this, we can set up the equation for the system's torque balance:

Torque due to beam's weight + Torque due to block's weight = Torque due to rope's tension

This gives the equation:

Substituting , we get:

Canceling out the common factors of and , we get:

Simplifying this, we find:

Step 4: Conclusion

The rope will break when . This is because when exceeds this value, the torque generated by the block becomes too large for the rope to handle, and it reaches its maximum tension threshold.

Final Answer:

The rope breaks if .

Given:

• A tube is inclined at angle θ = 45°
• The tube is accelerating down the incline with constant acceleration
• The tube is filled with water, and pressures at points 1 and 2 are and
• The expression to evaluate is:

Step 1: Consider effective acceleration on the fluid

Since the tube is accelerating along the incline, we analyze the effective gravity (pseudo-force method) in the non-inertial frame.

Effective acceleration on fluid = vector sum of actual gravity downward and pseudo-acceleration up the incline.

Break both vectors into components perpendicular and parallel to the incline. For incline angle :
- Gravity components:

- Pseudo-acceleration (opposing ) is entirely along incline:

Step 2: Effective acceleration along the tube

Effective component along the tube:

Step 3: Hydrostatic pressure difference due to effective gravity

Pressure difference in fluid column of height is: Use:

Step 4: Substitute

Simplify using: Hence: After simplifying:

Final Answer: when

Step 1: Understanding the System
We have a horizontal uniform beam of weight and length , hinged at point . The other end of the beam is supported by a light, inextensible rope that is fixed at point , at height above the hinge at point . A block of weight is attached at point of the beam.
The rope can sustain a maximum tension of . Our goal is to determine the reaction forces at point and the conditions under which the rope will break.

Step 2: Vertical Component of Reaction Force at
We will first analyze the forces acting on the system. Consider the vertical forces acting on the beam. The forces acting on the beam are:
- The weight of the beam, , acting vertically downward at the center of mass of the beam, i.e., at from the hinge . - The weight of the block, , acting vertically downward at point , which is a distance from the hinge . - The vertical component of the tension in the rope, denoted by , which acts vertically upward.
For vertical force equilibrium, we sum the forces acting vertically and set them equal to zero: Thus, the vertical reaction at point is independent of , as it only depends on the weight of the beam and the block.
Therefore, the correct statement is:
(A) The vertical component of reaction force at does not depend on .

Step 3: Horizontal Component of Reaction Force at
Now let's analyze the horizontal forces. The horizontal force acting at point is due to the tension in the rope, specifically the horizontal component of the tension force, denoted as .
The horizontal equilibrium condition gives: This condition holds true when , as the forces are balanced when the weight of the beam equals the horizontal tension component.
Therefore, the correct statement is:
(B) The horizontal component of reaction force at is equal to for .

Step 4: Condition for Rope Breaking
The maximum tension the rope can withstand is given as . The total tension in the rope depends on the forces acting on the beam and the geometry of the system.
We calculate the total tension in the rope by summing the forces and taking into account the angles involved in the system. Using the torque equilibrium condition around point , we find that the rope will break when the tension exceeds the maximum allowed value.
The rope will break when: This is the condition when the total tension in the rope exceeds the maximum allowed value.
Therefore, the correct statement is:
(D) The rope breaks if .

Final Answer:
The correct answers are:
(A) The vertical component of reaction force at does not depend on
(B) The horizontal component of reaction force at is equal to for
(D) The rope breaks if

Pressure-Volume Relation and Height Calculation

Given that is constant:

From this, we get the relation between pressure and volume:

If the volume changes by a factor of , the pressure changes by a factor of . Thus, the pressure becomes:

The equation for the pressure difference is:

Rearranging this to solve for :

Now, substituting the given values:

Solving for :