Twisting Torque | Twisting torque on a Cylinder or Wire | Twisting Torque | Angle of Twist | Mechanics | University Notes | Physics Notes | B.Sc. Physics Notes by Study Buddy Notes

Twisting torque on a Cylinder or Wire

Twisting Torque | Twisting torque on a Cylinder or Wire | Twisting Torque | Angle of Twist | Mechanics | University Notes | Physics Notes | B.Sc. Physics Notes by Study Buddy Notes

Introduction to Twisting Torque

• Definition: Twisting torque is the torque applied to an object in a way that causes it to rotate around its longitudinal axis. This type of torque is often applied to cylindrical objects like shafts or wires, creating a twisting or torsional effect.
• Concept of Torsion: Torsion refers to the twisting of an object due to an applied torque. When torque is applied, it creates an angular displacement and induces shear stress within the material.
• Applications: Twisting torque is critical in various fields, such as mechanical engineering and construction, and is especially relevant for components like drive shafts, springs, and power transmission lines, where materials are subject to torsional loads.

Understanding C

• Angular Displacement: When a cylinder or wire is twisted, it undergoes an angular displacement along its length. This is known as the angle of twist and depends on the applied torque, the material's rigidity, and the object's length.
• Shear Strain: The twisting action generates shear strain within the material, calculated as the ratio of deformation to the original length in the direction perpendicular to the applied force.
• Relation to Torque: Torsional deformation is directly related to the applied torque. Higher torque produces a greater angle of twist, assuming the material's properties and dimensions remain constant.

Shear Stress in Twisting Torque

• Definition of Shear Stress: In a twisted object, shear stress arises from internal forces that resist the applied torque. It varies along the radius, being zero at the center and maximum at the outer edge.
• Stress Distribution: Shear stress ($\tau$) at a given radius $r$ within a circular cross-section is given by: $$\tau = \frac{T \cdot r}{J}$$ where $T$ is the applied torque, $r$ is the radial distance from the center, and $J$ is the polar moment of inertia.
• Maximum Shear Stress: This occurs at the outer radius, where $r$ equals the radius of the cylinder or wire. Understanding maximum shear stress is crucial in material selection, as exceeding the shear strength of the material can lead to failure.

Torsion Formula and Derivation

The torsion formula provides a mathematical relationship for the behavior of a cylindrical object (such as a shaft or wire) under an applied twisting torque. This formula helps us calculate the relationship between torque, material properties, and the resulting angle of twist. Here’s the detailed derivation:

Consider a cylindrical shaft of:

• Length $L$,
• Radius $R$,
• Cross-sectional area $A$,
• Subjected to an applied torque $T$ at one end, while the other end is fixed.

This applied torque causes a twist in the shaft along its length, generating shear stress and shear strain within the material.

Definition of Shear Strain

Under the action of torque:

• The cylinder experiences an angular displacement, or twist, represented by the angle of twist $\theta$ (in radians).
• This results in a shear strain $\gamma$, which is defined as the angle (in radians) between two initially perpendicular lines on the cross-sectional area after deformation.

For a point at a distance $r$ from the axis, shear strain $\gamma$ is given by:

$$\gamma = \frac{r \theta}{L}$$

where:

• $r$ is the radial distance from the axis of the cylinder,
• $\theta$ is the angle of twist in radians over the entire length $L$.

Shear Stress and Shear Modulus

The shear stress $\tau$ at any point in the material is related to the shear strain $\gamma$ through the material’s shear modulus $G$, as:

$$\tau = G \cdot \gamma$$

Substitute $\gamma$ from the previous equation:

$$\tau = G \cdot \frac{r \theta}{L}$$

This equation shows that the shear stress $\tau$ is proportional to the radial distance $r$ from the center.

Torque and Polar Moment of Inertia

To find the total torque $T$ acting on the cross-section, we integrate the shear stress over the cross-sectional area:

$$T = \int_A \tau \cdot r \, dA$$

Substitute $\tau = G \cdot \frac{r \theta}{L}$

$$T = \int_A \left(G \frac{r \theta}{L}\right) \cdot r \, dA$$
$$T = \frac{G \theta}{L} \int_A r^2 \, dA$$

The term $\int_A r^2 \, dA$ is known as the polar moment of inertia $J$ of the cross-section:

$$J = \int_A r^2 \, dA$$

Therefore, the torque can be rewritten as:

$$T = \frac{G J \theta}{L}$$

Final Torsion Formula

Rearrange this equation to express the angle of twist $\theta$ in terms of the applied torque $T$:

$$\theta = \frac{T L}{G J}$$

This equation is the torsion formula, which relates:

• $T$: the applied torque,
• $G$: the shear modulus (a material property),
• $J$: the polar moment of inertia of the cross-section,
• $L$: the length of the cylinder,
• $\theta$: the angle of twist.

Summary of the Torsion Formula

The torsion formula is:

$$T = \frac{G J \theta}{L}$$

or equivalently,

$$\theta = \frac{T L}{G J}$$

• $T$: The applied torque, which causes the twisting.
• $G$: The shear modulus of the material, indicating its rigidity in shear.
• $J$: The polar moment of inertia, representing the resistance of the cross-section to twisting.
• $L$: The length of the shaft or cylinder, which affects the total deformation.
• $\theta$: The angle of twist, showing the extent of the angular displacement along the length.

Polar Moment of Inertia and its Importance

• Definition: The polar moment of inertia $J$ quantifies the resistance of a cross-section to twisting. It depends on the geometry of the cross-section, particularly its shape and size.
• Calculation for Circular Cross-Sections: For a solid cylinder of radius $R$, $J$ is given by: $$J = \frac{\pi R^4}{2}$$
• Effect on Torsion: A larger polar moment of inertia means greater resistance to twisting, so components with larger cross-sectional areas or hollow structures are often used for torsional applications.

Angle of Twist and its Calculation

The angle of twist ($\theta$) is the angular displacement experienced by a cylindrical object (such as a shaft or wire) when subjected to a twisting torque $T$. This angle depends on factors like the material's rigidity, the geometry of the cross-section, and the length of the object.

1. Definition of Angle of Twist

When a torque $T$ is applied to one end of a shaft, causing it to twist while the other end is fixed, the shaft experiences an angular displacement along its length $L$. This displacement, or rotation, is called the angle of twist and is typically measured in radians.

2. Formula for the Angle of Twist

The torsion formula gives us the relationship between the torque $T$, angle of twist $\theta$, and material and geometric properties:

$$T = \frac{G J \theta}{L}$$

where:

• $T$: Applied torque (Nm),
• $G$: Shear modulus of the material (Pa or N/m²),
• $J$: Polar moment of inertia of the cross-sectional area (m⁴),
• $\theta$: Angle of twist (in radians),
• $L$: Length of the shaft (m).

Rearrange this formula to solve for the angle of twist $\theta$:

$$\theta = \frac{T L}{G J}$$

3. Explanation of Each Term in the Formula

• $T$: The applied torque, which is the twisting force causing the rotation.
• $L$: The length of the shaft. Longer shafts will twist more under the same torque compared to shorter ones, assuming the same material and cross-section.
• $G$: Shear modulus, a property of the material that indicates its resistance to shear deformation. Materials with a higher $G$ value (like steel) are more resistant to twisting, resulting in a smaller angle of twist for the same applied torque.
• $J$: Polar moment of inertia, a geometric property of the cross-section that measures the resistance to twisting. A larger$J$ (such as in a thicker shaft) results in a smaller angle of twist for a given torque.

4. Calculation Steps

To calculate the angle of twist $\theta$, follow these steps:

1. Determine the Applied Torque $T$: Measure or know the torque applied to the shaft in Nm (Newton-meters).

2. Measure the Length $L$ of the Shaft: This is the length over which the torque is applied, typically in meters (m).

3. Find the Shear Modulus $G$ of the Material: Shear modulus depends on the material. For example, steel has a higher $G$ than aluminum, meaning it will twist less under the same conditions.

4. Calculate or Find the Polar Moment of Inertia $J$:

   • For a solid circular cross-section with radius $R$: $$J = \frac{\pi R^4}{2}$$
   • For a hollow circular cross-section with outer radius Ro​ and inner radius $R_i$: $$J = \frac{\pi (R_o^4 - R_i^4)}{2}$$

5. Apply the Torsion Formula:

   $$\theta = \frac{T L}{G J}$$

5. Units and Conversion

• Ensure that all units are consistent. Typically:
  • Torque $T$ in Nm,
  • Length $L$ in meters (m),
  • Shear modulus $G$ in Pascals (Pa or N/m²),
  • Polar moment of inertia $J$ in meters to the fourth power (m⁴).

The resulting angle of twist $\theta$ will be in radians. To convert it to degrees, multiply by $\frac{180}{\pi}$.

6. Example Calculation

Suppose we have a solid steel shaft with:

• Applied torque $T$ = 200 Nm,
• Length $L$ = 1.5 m,
• Radius $R$ = 0.05 m,
• Shear modulus $G$ for steel = $80 \times 10^9 \, \text{Pa}$.

1. Calculate the Polar Moment of Inertia $J$:

   $$J = \frac{\pi R^4}{2} = \frac{\pi (0.05)^4}{2} = 4.91 \times 10^{-6} \, \text{m}^4$$
   

2. Calculate the Angle of Twist $\theta$:

   $$\theta = \frac{T L}{G J} = \frac{200 \times 1.5}{(80 \times 10^9) \times 4.91 \times 10^{-6}}$$ $$\theta = \frac{300}{392.8} = 0.000764 \, \text{radians}$$
   

3. Convert $\theta$ to Degrees (if needed):

   $$\theta = 0.000764 \times \frac{180}{\pi} \approx 0.0438^\circ$$

So, the angle of twist for this steel shaft under the given conditions is approximately 0.000764 radians or 0.0438 degrees.

Shear Modulus (Modulus of Rigidity) and Material Properties

• Definition: The shear modulus, $G$, measures a material's resistance to shearing deformation. It is an intrinsic property of the material and depends on the atomic structure.
• Typical Values: Common materials have distinct shear moduli, e.g., steel (~80 GPa), aluminum (~26 GPa). Higher $G$ values indicate stiffer materials, which resist twisting more effectively.
• Effect on Torsional Resistance: Materials with a high shear modulus will exhibit less deformation under a given torque, making them suitable for high-stress applications like drive shafts.

Torsional Rigidity and Stiffness

• Torsional Rigidity ($GJ$): This product reflects the resistance of an object to twisting. Higher torsional rigidity means less deformation under applied torque.
• Calculation: Torsional stiffness $k$ is defined as the torque per unit angle of twist: $$k = \frac{G J}{L}$$
• Applications: Torsional rigidity is a crucial factor in the design of components like power transmission shafts, which must maintain stiffness to prevent excessive angular displacement under load.

Torsional Stress Distribution in a Cylinder or Wire

• Stress Variation: Shear stress varies linearly from the center to the edge of the cross-section, with maximum stress at the surface.
• Maximum Shear Stress: Given by: $$\tau_{\text{max}} = \frac{T R}{J}$$ where $R$ is the outer radius.
• Stress Graphs: Graphs can visually illustrate how stress varies across the radius, helping in understanding material behavior and failure points.

Twisting Failure and Yield Criteria

• Types of Failure: Under excessive torque, materials can yield (permanent deformation) or fracture.
• Yield Criteria: Maximum shear stress and von Mises criteria are commonly used to predict failure. If the applied shear stress exceeds the material's yield strength, deformation becomes irreversible.
• Ductile vs. Brittle Failure: Ductile materials undergo plastic deformation before failing, while brittle materials tend to fracture suddenly. Selection of material type is based on expected torsional loads.

Experimental Methods for Measuring Twisting Torque

• Torsion Testing Machines: Equipment applies controlled torque to samples, and sensors measure angular displacement and resistance.
• Procedure: A sample is clamped and twisted; measurements of torque and twist are recorded to calculate shear stress and modulus.
• Data Analysis: Results help in verifying theoretical models and ensuring material suitability for torsional applications.