A collar of negligible size slides over the surface of horizontal circular rod for which coefficient of friction is 0.3.The collar is projected with a speed of 4 m/s and the radius of the ring is 10 cm.Find the distance moved by the collar before coming to rest.g=9.8m/s²

To solve the problem of how far the collar moves before coming to rest, we can apply the principles of physics, particularly the concepts of friction and motion. Let's break it down step by step.

Understanding the Forces at Play

When the collar is projected along the circular rod, it experiences two main forces: the gravitational force acting downwards and the frictional force acting opposite to the direction of motion. The frictional force is what will eventually bring the collar to a stop.

Calculating the Frictional Force

The frictional force can be calculated using the formula:

• F_friction = μ * N

Where:

• μ is the coefficient of friction (0.3 in this case).
• N is the normal force. For a horizontal surface, this is equal to the weight of the collar, which can be expressed as N = m * g.

Since we don't have the mass of the collar, we can keep it as m for now.

Finding the Normal Force

Using the gravitational acceleration:

• g = 9.8 m/s²

The normal force becomes:

• N = m * 9.8

Substituting this into the frictional force equation gives:

• F_friction = 0.3 * m * 9.8 = 2.94m

Applying Newton's Second Law

According to Newton's second law, the net force acting on the collar will equal the mass times its acceleration:

• F_net = m * a

Since the frictional force is the only force acting to decelerate the collar, we can set up the equation:

• -F_friction = m * a

Substituting the expression for frictional force:

• -2.94m = m * a

We can cancel out the mass m (assuming it is not zero), leading to:

• a = -2.94 m/s²

Using Kinematic Equations

Now that we have the acceleration, we can use the kinematic equation to find the distance the collar travels before coming to rest:

• v² = u² + 2a s

Where:

• v is the final velocity (0 m/s when it comes to rest),
• u is the initial velocity (4 m/s),
• a is the acceleration (-2.94 m/s²),
• s is the distance traveled.

Plugging in the values:

• 0 = (4)² + 2(-2.94)s

This simplifies to:

• 0 = 16 - 5.88s

Rearranging gives:

• 5.88s = 16

Solving for s:

• s = 16 / 5.88 ≈ 2.72 m

Final Result

The collar moves approximately 2.72 meters before coming to rest. This calculation illustrates how friction plays a crucial role in decelerating moving objects, and it highlights the application of Newton's laws and kinematic equations in solving real-world physics problems.