Starting From Rest The Time Taken By A Body To Slide Down A 45° Inclined Plane With Friction Is Twice The Time Taken To Slide Down The Same Distance In Absence Of Friction. Determine The Coefficient Of Friction Between The Body And The Inclined Plane

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Introduction

Friction is a fundamental force that plays a crucial role in various physical phenomena, including the motion of objects on inclined planes. In this article, we will explore the relationship between friction and the time taken by a body to slide down a 45° inclined plane. We will also derive a formula to determine the coefficient of friction between the body and the inclined plane.

The Effect of Friction on Inclined Plane Motion

When a body is placed on an inclined plane, it experiences a force of gravity acting downward, which is resolved into two components: one parallel to the inclined plane and one perpendicular to it. The component parallel to the inclined plane is responsible for the body's motion down the plane. However, the presence of friction opposes this motion, causing the body to slow down and eventually come to rest.

The Role of Friction in Determining the Time of Descent

Let's consider a body of mass 'm' placed on a 45° inclined plane. In the absence of friction, the body will slide down the plane with a constant acceleration, given by:

a = g sin(θ)

where g is the acceleration due to gravity and θ is the angle of inclination.

The time taken by the body to slide down a distance 's' in the absence of friction is given by:

t = √(2s/g sin(θ))

Now, let's consider the case where friction is present. The force of friction opposing the motion of the body is given by:

F = μN

where μ is the coefficient of friction and N is the normal force acting on the body.

The net force acting on the body is the difference between the force of gravity and the force of friction:

F_net = mg sin(θ) - μN

Since the body is moving with a constant acceleration, we can write:

F_net = ma

where a is the acceleration of the body.

Substituting the expression for F_net, we get:

mg sin(θ) - μN = ma

Simplifying the equation, we get:

a = g sin(θ) - μg cos(θ)

The time taken by the body to slide down a distance 's' in the presence of friction is given by:

t = √(2s/(g sin(θ) - μg cos(θ)))

Comparing the Times of Descent

We are given that the time taken by the body to slide down the inclined plane with friction is twice the time taken to slide down the same distance in the absence of friction. Mathematically, this can be expressed as:

t_friction = 2t_no_friction

Substituting the expressions for t_friction and t_no_friction, we get:

√(2s/(g sin(θ) - μg cos(θ))) = 2√(2s/g sin(θ))

Squaring both sides and simplifying, we get:

g sin(θ) - μg cos(θ) = 4g sin(θ)

Substituting the value of θ = 45°, we get:

g/√2 - μg/√2 = 4g/√2

Simplifying the equation, we get:

μ = 3/2

Conclusion

In this article, we have derived a formula to determine the coefficient of friction between a body and an inclined plane. We have shown that the time taken by a body to slide down a 45° inclined plane with friction is twice the time taken to slide down the same distance in the absence of friction. By using this relationship, we have determined the coefficient of friction to be 3/2.

References

  • [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
  • [2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Glossary

  • Coefficient of friction: A dimensionless quantity that represents the ratio of the force of friction to the normal force acting on an object.
  • Inclined plane: A surface that is tilted at an angle to the horizontal.
  • Friction: A force that opposes the motion of an object when it is in contact with another surface.
  • Acceleration: The rate of change of velocity of an object with respect to time.

Q1: What is the effect of friction on inclined plane motion?

A1: Friction opposes the motion of an object on an inclined plane, causing it to slow down and eventually come to rest. The force of friction depends on the coefficient of friction between the object and the inclined plane.

Q2: How does the presence of friction affect the time of descent on an inclined plane?

A2: The presence of friction increases the time of descent on an inclined plane. In the absence of friction, the object will slide down the plane with a constant acceleration, while in the presence of friction, the object will experience a deceleration due to the force of friction.

Q3: What is the relationship between the time of descent and the coefficient of friction?

A3: The time of descent is inversely proportional to the coefficient of friction. As the coefficient of friction increases, the time of descent decreases.

Q4: How can we determine the coefficient of friction between an object and an inclined plane?

A4: We can determine the coefficient of friction by measuring the time of descent on an inclined plane with and without friction. By using the relationship between the time of descent and the coefficient of friction, we can calculate the coefficient of friction.

Q5: What is the significance of the 45° angle in inclined plane motion?

A5: The 45° angle is significant because it allows us to simplify the calculations and derive a general expression for the coefficient of friction. At this angle, the force of gravity acting on the object is resolved into two equal components, one parallel to the inclined plane and one perpendicular to it.

Q6: Can we apply the results of this analysis to other angles of inclination?

A6: Yes, we can apply the results of this analysis to other angles of inclination. However, the calculations will become more complex, and we will need to use trigonometric functions to resolve the force of gravity into its components.

Q7: What are the limitations of this analysis?

A7: The limitations of this analysis are that it assumes a constant coefficient of friction and neglects other forces that may be acting on the object, such as air resistance.

Q8: Can we use this analysis to design a system that minimizes friction and maximizes efficiency?

A8: Yes, we can use this analysis to design a system that minimizes friction and maximizes efficiency. By understanding the relationship between friction and inclined plane motion, we can design systems that take advantage of this relationship to achieve optimal performance.

Q9: What are some real-world applications of this analysis?

A9: Some real-world applications of this analysis include the design of conveyor belts, escalators, and other mechanical systems that involve inclined planes. By understanding the relationship between friction and inclined plane motion, we can design systems that are more efficient and effective.

Q10: Can we use this analysis to study other types of motion, such as rotational motion?

A10: Yes, we can use this analysis to study other types of motion, such as rotational motion. By applying the principles of friction and inclined plane motion to rotational motion, we can gain a deeper understanding of the underlying physics and develop new insights into the behavior of complex systems.

Conclusion

In this article, we have answered some frequently asked questions about friction and inclined plane motion. We have discussed the effect of friction on inclined plane motion, the relationship between the time of descent and the coefficient of friction, and the significance of the 45° angle. We have also explored some real-world applications of this analysis and discussed its limitations. By understanding the relationship between friction and inclined plane motion, we can design systems that are more efficient and effective.