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Chapter 8: Problem 72

Does the tip of a rotating fan blade have a tangential acceleration when the blade is rotating (a) at a constant angular velocity and (b) at a constant angular acceleration? Provide reasons for your answers. Problem A fan blade is rotating with a constant angular acceleration of \(+12.0 \mathrm{rad} / \mathrm{s}^{2}\). At what point on the blade, measured from the axis of rotation, does the magnitude of the tangential acceleration equal that of the acceleration due to gravity?

Short Answer

Expert verified
0.817 m from the axis of rotation; no tangential acceleration at constant angular velocity.

Step by step solution

01

Understand Tangential Acceleration

Tangential acceleration refers to the rate of change of the tangential velocity of a point on a rotating object. It occurs when there is a change in the angular velocity of the object.
02

Analyzing Case (a) - Constant Angular Velocity

When a rotating fan blade has a constant angular velocity, it means that the angular velocity is not changing with time. Therefore, the tangential acceleration, which depends on the change in angular velocity, is zero.
03

Analyzing Case (b) - Constant Angular Acceleration

When a blade rotates with a constant angular acceleration, it means the angular velocity is changing at a steady rate. Therefore, the tangential acceleration is non-zero and can be calculated using the relation between tangential acceleration and angular acceleration.
04

Relation between Tangential Acceleration and Angular Acceleration

For a point on a rotating object, the tangential acceleration \( a_t \) is given by \( a_t = \alpha \cdot r \), where \( \alpha \) is the angular acceleration and \( r \) is the distance from the axis of rotation.
05

Compare with Acceleration due to Gravity

We need the point where the tangential acceleration \( a_t \) is equal to the acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \). Substitute \( \alpha = 12.0 \, \text{rad/s}^2 \) and \( a_t = g \) into the formula \( a_t = \alpha \cdot r \).
06

Solve for the Distance from the Axis of Rotation

Setting \( \alpha \cdot r = g \), we have \( 12.0 \, \text{rad/s}^2 \cdot r = 9.8 \, \text{m/s}^2 \). Solve for \( r \):\[ r = \frac{9.8}{12.0} \approx 0.817 \text{ m} \]
07

Final Conclusion

The tip of the fan blade has no tangential acceleration when the blade rotates at a constant angular velocity, but it does when it rotates with a constant angular acceleration. The point where the tangential acceleration equals the acceleration due to gravity is approximately 0.817 meters from the axis of rotation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Velocity
Angular velocity is a measure of how fast an object rotates around a point or axis. Think of it as the speed at which the object is spinning. It is usually expressed in radians per second (rad/s).
  • Constant angular velocity means the speed of rotation doesn't change over time.
  • If a fan blade is spinning with constant angular velocity, each point on the blade moves in a circle at a steady rate.
  • This consistency in speed leads to no change in tangential velocity, which means the tangential acceleration is zero.
Understanding angular velocity helps explain why a fan blade rotating at constant speed doesn鈥檛 have varying speeds along its path.
Angular Acceleration
Angular acceleration is the rate at which angular velocity changes with time. In simpler terms, it's how quickly an object speeds up or slows down its spin. It is usually measured in radians per second squared (rad/s虏).
  • A constant angular acceleration implies that the object's spin speed increases or decreases steadily.
  • If a fan blade experiences angular acceleration, its angular velocity changes.
  • This change results in tangential acceleration as well, because the speed along the blade鈥檚 edge shifts.
When a fan blade has a constant angular acceleration, it鈥檚 continuously changing its angular velocity and thus its tangential velocity, causing non-zero tangential acceleration.
Centripetal Acceleration
Centripetal acceleration is the acceleration experienced by an object moving in a circular path. This type of acceleration points toward the center of the circle, keeping the object in its curved path.
  • Despite the name, centripetal acceleration does not change the speed of the object along its path, only the direction.
  • The formula for centripetal acceleration is given by \( a_c = \frac{v^2}{r} \), where \( v \) is the tangential speed and \( r \) is the radius of the circle.
  • Even with constant angular velocity, centripetal acceleration can be present because the direction of movement constantly changes.
Centripetal acceleration is essential for understanding how objects stay on their circular paths without flying off.
Equation of Motion
The equations of motion describe how an object's movement changes over time. They can be used to calculate positions, speeds, and accelerations in both linear and rotational contexts.
  • For rotational motion, there are counterparts to linear motion equations like \( v = u + at \), which becomes \( \omega = \omega_0 + \alpha t \) in rotational dynamics, where \( \omega \) is angular velocity and \( \alpha \) is angular acceleration.
  • These equations can calculate how quickly an object accelerates as it rotates.
  • Knowing these relationships helps in solving problems where different types of motion are involved, like a fan's blade rotation.
Applying equations of motion allows for precise calculation of changes in both rotational velocity and position.
Rotational Dynamics
Rotational dynamics is the study of how forces impact the rotation of objects. It involves understanding the effect of torque and rotational inertia on motion.
  • Just as linear dynamics involves forces acting in straight lines, rotational dynamics deals with forces causing objects to spin.
  • Torque, the rotational equivalent of force, causes changes in rotational speed and direction.
  • The distribution of mass around the axis (rotational inertia) affects how an object accelerates when a torque is applied.
Integrating concepts like torque, angular velocity, and angular acceleration allows us to predict and analyze how objects rotate. This forms the foundation for how machines like fans work efficaciously.

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Most popular questions from this chapter

A ball of radius \(0.200 \mathrm{~m}\) rolls along a horizontal table top with a constant linear speed of \(3.60 \mathrm{~m} / \mathrm{s} .\) The ball rolls off the edge and falls a vertical distance of \(2.10 \mathrm{~m}\) before hitting the floor. What is the angular displacement of the ball while the ball is in the air?

A CD has a playing time of 74 minutes. When the music starts, the \(\mathrm{CD}\) is rotating at an angular speed of 480 revolutions per minute (rpm). At the end of the music, the \(\mathrm{CD}\) is rotating at \(210 \mathrm{rpm}\). Find the magnitude of the average angular acceleration of the \(\mathrm{CD}\). Express your answer in \(\mathrm{rad} / \mathrm{s}^{2}\)

In the table are listed the initial angular velocity \(\omega_{0}\) and the angular acceleration \(\alpha\) of four rotating objects at a given instant in time. $$ \begin{array}{|c|c|c|} \hline & \text { Initial angular velocity } \omega_{0} & \text { Angular acceleration } \alpha \\ \hline(\mathrm{a}) & +12 \mathrm{rad} / \mathrm{s} & +3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline(\mathrm{b}) & +12 \mathrm{rad} / \mathrm{s} & -3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline(\mathrm{c}) & -12 \mathrm{rad} / \mathrm{s} & +3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline(\mathrm{d}) & -12 \mathrm{rad} / \mathrm{s} & -3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline \end{array} $$ In each case, state whether the angular speed of the object is increasing or decreasing in time. Account for your answers. Problem For each of the four pairs in the table, determine the final angular speed of the object if the elapsed time is \(2.0 \mathrm{~s}\). Compare your final angular speeds with the initial angular speeds and make sure that your answers are consistent with your answers to the Concept Question.

A spinning wheel on a fireworks display is initially rotating in a counterclockwise direction. The wheel has an angular acceleration of \(-4.00 \mathrm{rad} / \mathrm{s}^{2}\). Because of this acceleration, the angular velocity of the wheel changes from its initial value to a final value of \(-25.0 \mathrm{rad} / \mathrm{s}\). While this change occurs, the angular displacement of the wheel is zero. (Note the similarity to that of a ball being thrown vertically upward, coming to a momentary halt, and then falling downward to its initial position.) Find the time required for the change in the angular velocity to occur.

After \(10.0 \mathrm{~s},\) a spinning roulette wheel at a casino has slowed down to an angular velocity of \(+1.88 \mathrm{rad} / \mathrm{s}\). During this time, the wheel has an angular acceleration of -5.04 \(\mathrm{rad} / \mathrm{s}^{2}\). Determine the angular displacement of the wheel.
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