Question

A \(2 \mathrm{~m}\) long rod of radius \(1 \mathrm{~cm}\) which is fixed from one end is given a twist of \(0.8\) radians. What will be the shear strain developed ? (A) \(0.002\) (B) \(0.004\) (C) \(0.008\) (D) \(0.016\)

Short answer

(A) $0.002$ (B) $0.004$ (C) $0.008$ (D) $0.016$ Shear strain = (twist angle * (length / radius)) Shear strain = (0.8 radians) * (2 m / 0.01 m) = 0.8 * 200 = 160 None of the answer choices match the calculated shear strain of 160. Verify the exercise and answer choices for correctness.

Step-by-step solution

Convert the units

Before we can use the formula, we need to make sure all units are consistent. The radius is given in centimeters, while the length is in meters. So, let's convert the radius to meters: 1 cm = 0.01 m

Apply the formula

Now, we can use the formula for shear strain by plugging in the given values: Shear strain = (twist angle * (length / radius)) Shear strain = (0.8 radians) * (2 m / 0.01 m)

Calculate the shear strain

Perform the calculation to find the shear strain: Shear strain = (0.8 radians) * (2 m / 0.01 m) = 0.8 * 200 = 160

Compare with answer choices

Since none of the answer choices match the calculated shear strain of 160, there may be an error in the provided exercise or answer choices. Please verify the exercise and answer choices, and make sure they are correct.

Key concepts

Twist Angle Analysis

Twist angle refers to the measure of rotation applied to an object, particularly in rotational mechanics. In our problem, a twist angle of 0.8 radians is applied to a rod.
Understanding the twist involves recognizing that this angle indicates how much the end of the rod has rotated relative to its fixed point. Since radians are unitless but essential in calculating related quantities like shear strain, it's crucial to ensure the correctness of this value for accurate calculations.

When dealing with twist angles:

• Remember that they define angular displacement.
• Radians relate angles directly to the radius of the circle or arc involved.
• Maintaining unit consistency helps in accurately converting angular data to linear measures.

The twist angle is the beginning point in analyzing how much shear strain develops in materials like rods, which ultimately connects to their deformation under applied forces.

Radius Conversion Concerns

To solve practical problems involving rods or other cylindrical objects, it is critically important to ensure consistency in measurement units.
In this problem, while the rod's length is provided in meters, its radius is initially in centimeters. Ambiguities like this require conversion to maintain uniformity in units for calculations.

For conversions:

• Know that 1 cm equals 0.01 m.
• Convert the radius by multiplying with this factor to meet unit consistency.
• Applying conversions accurately prevents errors during the calculation of other parameters like shear strain.

By converting the 1 cm radius to 0.01 m, it ensures that all parameters in the problem are uniformly expressed, facilitating logical mathematical operations and reducing the scope for mistakes in engineering and physics calculations.

Ensuring Length Consistency

In physics and engineering contexts, maintaining consistency of length units across all components of a problem is crucial.
Our problem involves a rod that is 2 meters long, so it's vital that any other length-based values, like the radius or twist angle, are converted or calculated in relation to meters.

When working with length consistency:

• Always verify that all dimensions are in the same unit before performing calculations.
• Understand the relationship each measure has with the calculation, such as how length inversely impacts shear strain when measured correctly.
• Convert any inconsistent units before employing formulas involving these parameters.

Ensuring consistency of length units simplifies calculations and helps avoid errors, particularly when dealing with formulas dependent on accurate length measures, such as the formula for shear strain.