A block of mass m is placed on a smooth rod of mass m .Rod is horizontal initially. Two threads holding the block inclined at 45degree.Then the right thread is burnt. Just after burning of thread find Tension of left string?
To find the tension in the left string just after the right thread is burnt, we need to consider the forces acting on the block and the rod.
1. Weight of the block (mg): The weight of the block acts vertically downwards.
2. Tension in the left thread (Tl): The tension in the left thread acts at an angle of 45 degrees upwards.
3. Normal force by the rod (N): The normal force by the rod acts perpendicular to the rod's surface.
4. Tension in the right thread (Tr): The tension in the right thread acts horizontally towards the right.
Since the rod is smooth, there is no friction or torque acting on it.
Now, let's analyze the forces acting on the block along the vertical and horizontal axes.
Along the vertical axis:
- The weight of the block (mg) acts downwards.
- The vertical component of the tension in the left thread (Tl * sin 45°) acts upwards.
- The normal force (N) opposes the weight of the block.
So, we can write the equation for vertical forces as:
mg - Tl * sin 45° = N
Along the horizontal axis:
- The horizontal component of the tension in the left thread (Tl * cos 45°) acts towards the left.
- The tension in the right thread (Tr) acts towards the right.
Since the rod is smooth, there is no friction, so we can write the equation for horizontal forces as:
Tr - Tl * cos 45° = 0
Now, the crucial point is that just after the right thread is burnt, the forces acting on the block do not change. The only thing that changes is that the tension in the right thread (Tr) becomes zero.
So, after burning the right thread, the equation for horizontal forces becomes:
0 - Tl * cos 45° = 0
Simplifying the equation:
-Tl * cos 45° = 0
Solving for Tl:
Tl = 0 / cos 45°
Tl = 0 Newtons
Therefore, the tension in the left string just after burning the right thread is zero.
To find the tension in the left string just after the right thread is burnt, we can use the principle of angular momentum conservation.
Here's how you can approach the problem step by step:
1. Determine the initial angular momentum: The system initially consists of the block and the rod, which have rotational motion around their center of mass. Since the rod is horizontal initially, the initial angular momentum of the system is zero.
2. Burn the right thread: When the right thread is burnt, the block and the rod are no longer connected. The block will start to fall due to gravity, while the rod will rotate about its center of mass.
3. Apply the principle of angular momentum conservation: Just after the right thread is burnt, the block and the rod are still in contact, so their angular momenta will be conserved around their common center of mass.
Let's assume the tension in the left string is T (which is what we want to find).
4. Write the equation for angular momentum conservation: The initial angular momentum is zero, and it remains zero just after the right thread is burnt. The equation can be written as:
(m * g * l * sin(45°)) - (T * l) = 0
Here, m is the mass of the block, g is the acceleration due to gravity, and l is the length of the rod.
5. Solve the equation for T: Rearrange the equation and solve for T:
T = m * g * sin(45°)
Therefore, the tension in the left string just after the right thread is burnt is T = m * g * sin(45°).
Note: The assumption here is that there is no friction between the block and the rod, and the rod is a thin rod with a negligible mass compared to the block.