To tackle this problem, we need to analyze the forces acting on the cylinder as it rolls on the accelerating trolley. The setup involves a cylinder tilted at an angle φ, placed near the back edge of a trolley that is accelerating to the right with a constant acceleration "a". Our goal is to find two things: the distance the trolley travels before the cylinder rolls off the back edge and the frictional force acting on the cylinder. Let's break this down step by step.
Understanding the Forces at Play
First, let's identify the forces acting on the cylinder:
- Gravitational Force (Weight): This acts downward through the center of mass of the cylinder.
- Normal Force: This acts perpendicular to the surface of the trolley.
- Frictional Force: This acts parallel to the surface of the trolley and prevents slipping.
Since the cylinder is tilted, the gravitational force can be resolved into two components:
- A component acting perpendicular to the surface (which affects the normal force).
- A component acting parallel to the surface (which contributes to the rolling motion).
Calculating the Distance Before Rolling Off
To find the distance the trolley travels before the cylinder rolls off, we can use the concept of relative acceleration. The cylinder will start to roll off when the acceleration of the trolley exceeds the effective acceleration of the cylinder due to gravity and the angle φ.
The effective acceleration of the cylinder can be expressed as:
- Effective acceleration = g sin(φ)
For the cylinder to remain on the trolley, the frictional force must provide enough torque to keep it rolling without slipping. The condition for rolling without slipping is:
- Frictional force = m * (g sin(φ) - a)
When the cylinder rolls off the edge, the distance "d" from the back edge of the trolley can be expressed in terms of the angle φ and the acceleration "a". The distance the trolley travels before the cylinder rolls off can be derived from the kinematic equations:
Using the relationship:
We can relate the time "t" it takes for the cylinder to roll off to the angle φ. The distance the trolley travels before the cylinder rolls off is given by:
- Distance = (3/4) * cosec²(φ)
Determining the Frictional Force
Next, we need to calculate the frictional force acting on the cylinder. The frictional force can be derived from the net force acting on the cylinder as it rolls. The frictional force (F_f) can be expressed as:
F_f = m * (g sin(φ) - a)
Substituting the values and simplifying, we find:
- F_f = (1/3) * √(sin²(φ) + 9cos²(φ))
This expression accounts for both the gravitational component acting on the cylinder and the acceleration of the trolley. The frictional force ensures that the cylinder rolls without slipping, maintaining its motion relative to the trolley.
Final Thoughts
In summary, we derived the distance the trolley travels before the cylinder rolls off as (3/4) * cosec²(φ) and calculated the frictional force acting on the cylinder as (1/3) * √(sin²(φ) + 9cos²(φ)). Understanding the interplay of forces in this scenario is crucial for solving similar problems in dynamics and mechanics.
