Question

A force of \(8.8 \mathrm{~N}\) pushes on the rim of a wheel of radius \(0.41 \mathrm{~m}\). The direction of the force is at an angle of \(22^{\circ}\) relative to the radial direction. What is the torque produced by this force?

Short answer

The torque produced is approximately 1.36 Nm.

Step-by-step solution

Understanding the Definition of Torque

Torque is defined as the product of the force applied and the radius of the wheel, and it's influenced by the angle between the force and the radius. The formula for torque \(\tau\) is \(\tau = r \cdot F \cdot \sin(\theta)\), where \(r\) is the radius, \(F\) is the magnitude of the force, and \(\theta\) is the angle between the force and the radial direction.

Identifying Known Values

From the problem, we have \(F = 8.8 \mathrm{~N}\), \(r = 0.41 \mathrm{~m}\), and \(\theta = 22^{\circ}\). These values will be used in the torque formula.

Converting the Angle from Degrees to Radians

The angle \(22^{\circ}\) must be converted to radians for calculation purposes. The conversion is done using \(\theta \text{ in radians} = \theta \text{ in degrees} \times \frac{\pi}{180}\). So, \(\theta = 22 \times \frac{\pi}{180}\).

Calculating the Sine of the Angle

Calculate \(\sin(\theta)\) where \(\theta = 22^{\circ}\) converted to radians. Use a calculator to find \(\sin(22 \times \frac{\pi}{180})\).

Applying the Torque Formula

Using the torque formula \(\tau = r \cdot F \cdot \sin(\theta)\), plug in \(r = 0.41\), \(F = 8.8\), and the calculated \(\sin(\theta)\). Compute \(\tau\).

Calculating the Final Result

Compute \(\tau = 0.41 \times 8.8 \times \sin(22 \times \frac{\pi}{180})\). After computing, the torque \(\tau\) is approximately \(1.36 \mathrm{~Nm}\).

Key concepts

Understanding Force Vector

A force vector is a way to represent the direction and magnitude of a force acting on an object. In this problem, the force applied is represented as a vector with a magnitude of 8.8 N, pointing at an angle of 22 degrees.
This angle is significant because it determines how much of the force contributes to the rotational effect, or torque, on the wheel.

• Magnitude: This is simply the 'strength' of the force, measured in newtons (N).
• Direction: The angle relative to a reference line, like the radial direction of the wheel.

Force vectors are essential in physics as they allow us to calculate the effect of the force, particularly when it causes rotational motion.

Angle Conversion Made Easy

In physics problems, angles are often measured in degrees but need to be converted to radians for calculations involving trigonometric functions. Radians are the standard unit for these calculations:
To convert degrees to radians, we multiply the angle in degrees by \( \frac{\pi}{180} \).
For this task:

• Start with the angle: 22 degrees.
• Use the formula: \( \text{radians} = 22 \times \frac{\pi}{180} \).
• Calculate: This gives the radian measure that is used in further calculations.

Understanding this conversion is vital because most scientific calculations default to radian measure for accuracy and consistency.

The Role of the Sine Function

The sine function is crucial in determining the component of the force that contributes to torque. It lets us calculate the perpendicular portion of the force relative to the radius:

• The formula for torque includes \( \sin(\theta) \), showing the sine's importance.
• By finding \( \sin(\theta) \), we reveal how much of the force is not aligned with the radius. This is the effective force for generating rotation.

For our example, after converting the angle to radians, calculate \( \sin(22 \times \frac{\pi}{180}) \) to determine the force's effective component in the torque calculation.
Make sure to always use a calculator with trigonometric function capabilities to ensure precision.

Applying Physics Problem Solving Techniques

Solving physics problems such as torque calculations relies on a structured approach: break down the problem into manageable steps. This ensures clarity and accuracy.

• Identify all known and unknown quantities. Here, the force, radius, and angle were given.
• Divide the problem into steps: Understanding concepts, making conversions, and applying formulas.
• Verify each step: Look at how the known values lead to the solution, like converting angles and calculating functions.

For this torque problem, the systematic approach led us to use the formula \( \tau = r \cdot F \cdot \sin(\theta) \) and compute the results efficiently. Proficiency in these problem-solving techniques is invaluable in tackling complex physics questions.