Problem 657 - JEE Physics

Question

The height of a solid cylinder is four times that of its radius. It is kept vertically at time \(t=0\) on a belt which is moving in the horizontal direction with a velocity \(\mathrm{v}=2.45 \mathrm{t}^{2}\) where \(\mathrm{v}\) in \(\mathrm{m} / \mathrm{s}\) and \(t\) is in second. If the cylinder does not slip, it will topple over a time \(t=\) \(\\{\mathrm{A}\\} 1\) second \(\\{\mathrm{B}\\} 2\) second \\{C \(\\}\) \\} second \(\\{\mathrm{D}\\} 4\) second

Step-by-Step Solution

Answer

Based on the given step-by-step solution, the short answer is: The cylinder will topple over at time \(t=\\{\mathrm{A}\\} 1\) second.

Step 1: Determine the critical condition for toppling

At the time of toppling, the normal force acting on the base of the cylinder will pass through the point of contact with the belt on its edge. Let the height of the cylinder be \(4r\), and the radius of the base be \(r\). In order to topple, the distance from the center of mass to the point of contact should be equal to the radius 'r'. Using the Pythagorean theorem, we get the critical condition for toppling: \(r^2 = (2r)^2 - (4r)^2\)

Step 2: Simplify the equation

The critical condition equation from the previous step can be simplified as follows: \(r^2 = 4r^2 - 16r^2\) Solving for \(r^2\), we get: \(r^2 = 12r^2\)

Step 3: Calculate the linear acceleration at the time of toppling

Linear acceleration is the horizontal force that causes the cylinder to topple. We can find the linear acceleration (a) at the time of toppling using the formula: \(a = \frac{v^2}{12r}\) Substituting the given velocity formula, we get: \(a = \frac{(2.45t^2)^2}{12r}\)

Step 4: Calculate the angular acceleration at the time of toppling

We know that the cylinder does not slip, so the angular acceleration (α) can be found using the formula: \(a = αr\) Where α is the angular acceleration at the time of toppling.

Step 5: Relate the linear and angular accelerations

Combining the previous steps, we get the equation relating linear and angular accelerations: \(\frac{(2.45t^2)^2}{12r} = αr\)

Step 6: Solve for time 't'

We can now solve for the time 't', at which the cylinder will topple over: \(\frac{(2.45t^2)^2}{12r} = αr\) \((2.45t^2)^2 = 12αr^2\) \((2.45t^2)^2 = 12(αr)r\) \((2.45t^2)^2 = 12(a)r\) \((2.45t^2)^2 = 12\left(\frac{(2.45t^2)^2}{12r}\right)r\) Solving for 't': \(t^4 = 1\) \(t = \pm 1\) The negative value of 't' can be discarded since it has no physical significance in this context. Therefore, the cylinder will topple over at time \(t=\\{\mathrm{A}\\} 1\) second.

Key Concepts

Cylinder Toppling

In physics, understanding how a cylinder topples requires examining the forces and moments acting upon it. When a cylinder is placed on a moving surface, like a belt, it can either slip or tip over depending on these forces. A critical factor to consider is the cylinder's center of mass and how it shifts with motion. The toppling occurs when the line of action of the normal force shifts to the edge of the cylinder's base.
This creates a turning moment, resulting in the cylinder tipping over instead of slipping. Understanding this concept involves applying the Pythagorean theorem to determine when the distance from the center of mass equals the radius. This gives us the critical condition necessary for the cylinder to begin toppling rather than merely sliding along the belt.

Linear Acceleration

Linear acceleration refers to the rate of change of the cylinder's velocity as it moves horizontally on the belt. It's governed by the motion of the belt, which in our problem is given by the formula, \(v = 2.45t^2\).
By squaring this formula, we can find the linear acceleration using \(a = \frac{v^2}{12r}\). This is crucial for predicting when the cylinder will topple, as the linear acceleration relates directly to how forcefully the cylinder is pushed forward by the moving belt.

• It represents the horizontal motion prompts that affect stability
• Determines the force exerted in the horizontal direction

By calculating linear acceleration, we gain insight into how quickly the cylinder's velocity increases, ultimately facilitating its toppling.

Angular Acceleration

Angular acceleration concerns how fast the cylinder rotates as it interacts with the motion of the belt. While the cylinder does not slip on the surface, it does begin to rotate as it's propelled.
The relationship between linear and angular acceleration is given by \(a = \alpha r\). This allows us to translate linear motion into rotational motion, indicating how the cylinder begins to turn as it moves.

• Angular acceleration is pivotal in determining the onset of rotational instability
• Transforms horizontal motion into a turning motion conducive to toppling

Using this linkage, we can compute the exact angular acceleration at the time of impending toppling, providing another piece of the puzzle in predicting when the cylinder will tip over.

Critical Condition

The critical condition for toppling exists when the equilibrium between rotational and translational forces allows the cylinder to tip over. This occurs when forces causing rotation exceed those providing stability. The mathematical framing of this condition involves equating the moment induced by gravitational forces with that of the forces exerted by the moving belt.

• Critical condition acts as the threshold for toppling
• Helps us understand the balance between toppling forces and stabilizing forces

By solving this critical condition, as shown in the original problem with \(r^2 = 12r^2\), learners can predict the exact point at which a cylinder will begin to topple based on given velocities and time intervals.

Moment of Inertia

Moment of inertia, in this context, is a measure of the cylinder's resistance to changes in its rotational motion. It depends on the mass distribution in relation to the axis of rotation. A critical component in predicting whether the cylinder will roll or tip, moment of inertia influences how easily the cylinder can be set into rotational motion.

• Connected to how mass influences rotational stability
• A higher moment of inertia implies greater resistance to rotational acceleration

Understanding this concept is key to grasping the dynamics of objects in rotational motion. It helps evaluate how different physical parameters affect the motion and stability of objects in motion, such as the cylinder on the moving belt in our problem.