Lesson 1.3: Modelling Assumptions: Particle, Light and Rigid Bodies

Introduction

In this lesson, we will explore the essential modelling assumptions in mechanics: the particle model, the light model, and the rigid-body model. These modelling techniques allow us to simplify complex real-world problems and analyze them mathematically. By understanding these models, you will be equipped to tackle a wide range of mechanics problems effectively.

Learning Objectives

By the end of this lesson, you will be able to:

• Describe the particle model and understand its assumptions.
• Explain the light model and its implications.
• Differentiate between a particle and an extended rigid body.
• State the simplifications provided by the particle and light assumptions.
• Identify when to use the particle model versus the rigid-body model.

The Particle Model

The particle model simplifies any object into a dimensionless point mass, which has mass but occupies no space. This assumption allows us to ignore aspects like size, rotation, and air resistance, focusing only on the motion of the object itself.

Key Assumptions of the Particle Model

1. Negligible Size: The object is considered to have no dimensions. This means we treat the object as a point in space.
2. Negligible Rotation: The effects of the object's rotation are ignored. Only linear motion is considered.
3. Negligible Air Resistance: Effects of air resistance acting on the particle are not considered unless stated otherwise.

When is the Particle Model Appropriate?

The particle model is suitable in situations where the dimensions and orientation of an object do not significantly affect the problem. It is commonly used in:

• Free-fall problems, where objects are subject to gravity only.
• Projectile motion, where the path can be approximated as a parabolic curve.

Worked Example: Projectile Motion

Problem: A ball is thrown upwards with an initial velocity of $u = 20 \, \text{m/s}$. Calculate the maximum height it reaches before falling back down. Assume there is no air resistance.

Solution:

1. Use the formula for maximum height in projectile motion:

$$H = \frac{u^2}{2g}$$

where $g = 9.81 \, \text{m/s}^2$ is the acceleration due to gravity.

1. Substituting the values:

$$H = \frac{(20)^2}{2 \times 9.81}$$

$$H = \frac{400}{19.62}$$

$$H \approx 20.39 \, \text{m}$$

1. Therefore, the maximum height reached by the ball is approximately $20.39 \, \text{m}$.

The Light Model

The light model considers certain objects, such as strings, rods, or pulleys, to have negligible mass. This simplification is significant in analyzing systems where the weight of these components does not affect the overall behavior of the system.

Key Assumptions of the Light Model

1. Negligible Mass: The mass of strings, rods, or pulleys is considered so small that it does not influence the calculation of forces.
2. Ideal Conditions: The model assumes no stretching or bending occurs in these light objects, and they transmit forces without loss.

When is the Light Model Appropriate?

The light model is useful in cases of tension analysis in static systems or in cases of pulleys where the mass of the supporting ropes can be ignored.

Worked Example: Tension in a Light String

Problem: In a system where two blocks of mass $m_1 = 3 \, \text{kg}$ and $m_2 = 5 \, \text{kg}$ are connected by a light string over a frictionless pulley, find the tension in the string when the system is released.

Solution:

1. Set up free-body diagrams for both masses.
2. For $m_1$, using Newton's second law:

$$T - m_1g = -m_1a$$

For $m_2$:

$$m_2g - T = m_2a$$

1. Adding both equations:

$$m_2g - m_1g = (m_1 + m_2)a$$

1. Solve for $a$ (acceleration):

$a = \frac{(m_2 - m_1)g}{m_1 + m_2}$

Substituting values:

$a = \frac{(5 - 3)(9.81)}{3 + 5}$

$$a = \frac{19.62}{8}$$

$$a \approx 2.45 \, \text{m/s}^2$$

1. Substituting $a$ back to find the tension:

$$T = m_1g + m_1a$$

$$T = 3(9.81) + 3(2.45)$$

$$T \approx 29.43 \, \text{N}$$

1. Thus, the tension in the string is approximately $29.43 \, \text{N}$.

The Rigid-Body Model

The rigid-body model treats an object as having a fixed shape and mass distribution. Unlike particles, rigid bodies can experience rotational motion and have moments of inertia that need to be considered in dynamic analysis.

Key Differences Between a Particle and a Rigid Body

1. Shape and Size: Whereas a particle is dimensionless, a rigid body has a defined size and shape.
2. Rotational Motion: Rigid bodies can rotate around an axis, making rotational dynamics relevant.
3. Mass Distribution: The effects of how mass is distributed affect the behavior of rigid bodies (e.g., moment of inertia).

Common Applications of the Rigid-Body Model

• Analyzing the motion of objects such as beams, vehicles, or structures under the influence of forces.
• Studying the effects of torque and angular acceleration on rotating bodies.

Worked Example: Torque Calculation on a Rigid Body

Problem: A door with a width of $0.8 \, \text{m}$ and a mass of $10 \, \text{kg}$ is pushed at the edge (furthest from the hinges) with a force of $50 \, \text{N}$. Calculate the torque ($\tau$) about the hinges.

Solution:

1. Torque is calculated as:

$$\tau = rF\sin(\theta)$$

where $r$ is the distance from the pivot point (in this case, the hinges), $F$ is the applied force, and $\theta$ is the angle between the force and the lever arm.

1. In this case, $r = 0.8 \, \text{m}$, $F = 50 \, \text{N}$, and $\theta = 90^\circ$ (the force is applied perpendicularly).
2. Thus, we have:

$$\tau = 0.8 \times 50 \times \sin(90^\circ)$$

$$\tau = 0.8 \times 50 \times 1$$

$$\tau = 40 \, \text{N}\cdot\text{m}$$

1. Therefore, the torque applied on the door about the hinges is $40 \, \text{N}\cdot\text{m}$.

Conclusion

In this lesson, we have explored the fundamental modelling assumptions of mechanics: the particle model, the light model, and the rigid-body model. These models simplify the complexity of real situations, allowing us to analyze motion and forces with clarity. Understanding when to use each model is paramount in solving mechanics problems correctly and effectively.

Study Notes

• Particle Model: Treats objects as dimensionless point masses.
• Light Model: Assumes objects like strings and rods have negligible mass.
• Rigid Body Model: Considers the fixed shape and mass distribution of an object.
• Application: Use the particle model for simple linear motion and the light model for systems involving tension in strings or rods.
• Differentiate: Recall differences between particles and rigid bodies, particularly concerning rotation and the significance of mass distribution.