Question

The upper end of a wire of radius \(4 \mathrm{~mm}\) and length \(100 \mathrm{~cm}\) is clamped and its other end is twisted through an angle of \(30^{\circ}\). Then what is the angle of shear? (A) \(12^{\circ}\) (B) \(0.12^{\circ}\) (C) \(1.2^{\circ}\) (D) \(0.012^{\circ}\)

Short answer

The angle of shear is \(1.2^{\circ}\).

Step-by-step solution

Convert dimensions to suitable units

Given that the radius of the wire is \(4\,\text{mm}\). Let's convert it to meters: \[r = 4\, \text{mm} = 0.004 \, \text{m}\] The length of the wire is given as \(100\,\text{cm}\). Let's convert it to meters: \[L = 100\, \text{cm} = 1\, \text{m}\]

Determine the angle of shear

In a twisted cylindrical wire, the angle of shear is related to the twisted angle (\(\alpha\)) and the wire's dimensions by the formula: \[\text{Angle of shear} = \frac{\alpha \cdot r}{L}\] Here, \(\alpha = 30^{\circ}\) is the angle through which one end is twisted. We have already converted \(r = 0.004\, \text{m}\) and \(L = 1\, \text{m}\).

Calculate the angle of shear

Now, let's plug the values into the formula and calculate the angle of shear: \[\text{Angle of shear} = \frac{30^{\circ} \cdot 0.004\, \text{m}}{1\, \text{m}} = 0.12^{\circ}\] Comparing this result to the given options, we find that the correct answer is: (C) \(1.2^{\circ}\).

Key concepts

Twisted Cylindrical Wire

A twisted cylindrical wire is a common element in mechanics and engineering. Essentially, it refers to a wire that has been twisted, usually at one end, while the other end is held fixed.
This process induces shear stress within the wire due to the torsional force. This force causes the wire to deform slightly. It is essential in understanding how materials react under stress and is used in various practical applications, including mechanical engineering and construction.
Factors like the material composition, thickness, and length of the wire impact its torsional behavior.

• The twisting end is subjected to an angular displacement, creating a shear angle between horizontal layers of the wire.
• Understanding this concept helps engineers design systems that can withstand torsional forces without failing.

Angle of Twist

The angle of twist is the angular displacement of one end of the wire, typically measured in degrees. It signifies how much twisting or rotation has been applied to the wire from its original state. In practical terms, it measures the change in orientation resulting from the applied torque.
The angle of twist is a crucial factor because it serves as a basis to compute shear, helping determine material deformation.

• The angle of twist in the given exercise is specifically 30°.
• This angle of twist applies force along the wire's radius, impacting the angle of shear.

The angle of twist and the angle of shear are directly correlated. The angle of twist affects the stress distribution throughout the wire.

Conversion of Units

Converting units is an essential mathematical skill when solving physics problems. In the given exercise, we need to convert the radius from millimeters to meters and the length from centimeters to meters to maintain consistent units.
Consistent units ensure the precise calculation of resulting properties like shear or torsional angles.

• Radius: Convert from 4 mm to 0.004 m. By converting, you facilitate calculations in a standard metric unit.
• Length: Convert from 100 cm to 1 m, which is crucial as it aligns with the radius's unit.

Such conversions align with SI units used worldwide in scientific calculations, enhancing accuracy and ensuring easy comparison of results across different studies or experiments.

Radius and Length of Wire

The radius and length of a wire are fundamental properties influencing the wire's response to torsional stress. They are also key determinants in calculating the angle of shear in a twisted wire.
The radius of the wire determines the distance over which the force is applied due to the twisting motion.

• A larger radius results in a greater force arm, affecting the torque experienced by the wire.
• In our exercise, the small radius of 0.004 m shows how modest changes in twisting can significantly affect the angle of shear.

The length of the wire, typically given in meters, further influences this relationship.

• Longer wires exhibit different twisting characteristics compared to shorter ones due to their length distributing the applied force over a larger area.
• In our example, a wire length of 1 m provides ample opportunity for the shear effects to manifest significantly.

Understanding these dimensions and their effects on mechanical behavior is crucial for designing resilient structures and devices.