Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

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 \(1.36 \mathrm{~Nm}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

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.
  • 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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The following angles are given in degrees. Convert them to radians: \(30^{\circ}, 45^{\circ}, 90^{\circ}, 180^{\circ}\).

Explain How does rolling motion differ from pure rotational motion? How does it differ from pure linear motion?

Dragonflies often dip into the water on the surface of a lake or pond, presumably to drink or to clean themselves. Recently, it has been discovered that after dipping the dragonflies fly straight upward, then pitch forward and tumble head over tail several times until the water has been shed from their bodies. Observations show that the dragonflies complete one revolution every \(0.017 \mathrm{~s}\) during this "spin dry" maneuver. What is their angular speed in radians per second and revolutions per minute? (For comparison, a typical car engine runs at about \(1500 \mathrm{rpm}\).)

Force to Hold a Baseball A person holds a \(1.42-\mathrm{N}\) baseball in his hand, a distance of \(34.0 \mathrm{~cm}\) from the elbow joint, as shown in Figure \(8.30\). The biceps, attached at a distance of \(2.75 \mathrm{~cm}\) from the elbow, exerts an upward force of \(12.6 \mathrm{~N}\) on the forearm. Consider the forearm and hand to be a uniform rod with a mass of \(1.20 \mathrm{~kg}\). (a) Calculate the net torque acting on the forearm and hand. Use the elbow joint as the axis of rotation. (b) If the net torque obtained in part (a) is nonzero, in which direction will the forearm and hand rotate?

Find the angular speed of the Earth as it orbits about the Sun. Give your answer in radians per second (rad/s).

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free