Chapter 8: Problem 33
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
Expert verified
The torque produced is approximately 1.36 Nm.
Step by step solution
01
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.
02
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.
03
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, \(\theta = 22 \times \frac{\pi}{180}\).
04
Calculating the Sine of the Angle
Calculate \(\sin(\theta)\) where \(\theta = 22^{\circ}\) converted to radians. Use a calculator to find \(\sin(22 \times \frac{\pi}{180})\).
05
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 \(\tau\).
06
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 \(1.36 \mathrm{~Nm}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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:
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.
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:
Make sure to always use a calculator with trigonometric function capabilities to ensure precision.
- 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.
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.