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Chapter 10: Problem 4

The angular position of a point on a rotating wheel is given by \(\theta=2.0+4.0 t^{2}+2.0 t^{3}\), where \(\theta\) is in radians and \(t\) is in seconds. At \(t=0\), what are (a) the point's angular position and (b) its angular velocity? (c) What is its angular velocity at \(t=4.0 \mathrm{~s}\) ? (d) Calculate its angular acceleration at \(t=2.0 \mathrm{~s}\). (e) Is its angular acceleration constant?

Short Answer

Expert verified
(a) 2.0 rad, (b) 0 rad/s, (c) 128.0 rad/s, (d) 32.0 rad/s虏, (e) No, it's not constant.

Step by step solution

01

Find Angular Position at t=0

To find the angular position at \( t = 0 \), substitute \( t = 0 \) into the equation \( \theta = 2.0 + 4.0t^2 + 2.0t^3 \).\[ \theta = 2.0 + 4.0(0)^2 + 2.0(0)^3 = 2.0 \] Thus, the angular position is \( 2.0 \) radians.
02

Derive Angular Velocity Formula

The angular velocity \( \omega \) is the derivative of the angular position \( \theta \) with respect to time \( t \). Differentiate \( \theta = 2.0 + 4.0t^2 + 2.0t^3 \):\[ \omega = \frac{d\theta}{dt} = 0 + 8.0t + 6.0t^2 = 8.0t + 6.0t^2 \]
03

Find Angular Velocity at t=0

Using the angular velocity formula derived in Step 2, substitute \( t = 0 \) to find the angular velocity:\[ \omega = 8.0(0) + 6.0(0)^2 = 0 \] Thus, the angular velocity at \( t = 0 \) is \( 0 \) radians per second.
04

Find Angular Velocity at t=4s

Substitute \( t = 4.0 \) seconds into the angular velocity formula:\[ \omega = 8.0(4.0) + 6.0(4.0)^2 = 32.0 + 96.0 = 128.0 \] Thus, the angular velocity at \( t = 4.0 \) seconds is \( 128.0 \) radians per second.
05

Derive Angular Acceleration Formula

The angular acceleration \( \alpha \) is the derivative of the angular velocity \( \omega \) with respect to time \( t \).Differentiate \( \omega = 8.0t + 6.0t^2 \):\[ \alpha = \frac{d\omega}{dt} = 8.0 + 12.0t \]
06

Calculate Angular Acceleration at t=2s

Using the formula derived in Step 5, substitute \( t = 2.0 \) seconds to find the angular acceleration:\[ \alpha = 8.0 + 12.0(2.0) = 8.0 + 24.0 = 32.0 \] Thus, the angular acceleration at \( t = 2.0 \) seconds is \( 32.0 \) radians per second squared.
07

Determine if Angular Acceleration is Constant

The angular acceleration formula is \( \alpha = 8.0 + 12.0t \), which is a linear function of \( t \). Since this formula includes a term involving \( t \), the angular acceleration is not constant over time.

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Key Concepts

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

Angular Position
Angular position describes the orientation of an object in a rotational motion about a fixed axis. It is denoted by the symbol \( \theta \) and is typically measured in radians. The angular position provides a measure of how far a point has rotated from a reference direction. This concept is analogous to linear position in translational motion but in a circular path.
For instance, if you have a wheel with markers indicating its position, the angular position tells where around the circle the marker is, in terms of radians. A full circle is \( 2\pi \) radians.
In our exercise, the angular position is given by the equation \( \theta=2.0+4.0t^{2}+2.0t^{3} \). To find the angular position at specific times, simply substitute the time \( t \) into the equation. For example, at \( t = 0 \), the position is \( 2.0 \) radians.
Angular Velocity
Angular velocity represents how fast an object is rotating. It is the rate of change of angular position with respect to time and is measured in radians per second. It provides insight into how quickly the angle (or rotation) is changing as time progresses.
To find angular velocity, you derive the angular position with respect to time \( t \). The derivative of the given equation \( \theta = 2.0 + 4.0t^2 + 2.0t^3 \) gives the angular velocity \( \omega = 8.0t + 6.0t^2 \).
  • At \( t = 0 \), the angular velocity is \( 0 \) radians per second, meaning initially, the object isn't rotating.
  • At \( t = 4 \) seconds, substituting into the derived formula gives an angular velocity of \( 128.0 \) radians per second, indicating a significant increase in rotation speed over time.
Angular Acceleration
Angular acceleration is the rate of change of angular velocity over time. It provides information about how the rotational speed of an object is changing. This is crucial to understanding the dynamics of rotational motion.
The angular acceleration \( \alpha \) is found by differentiating the angular velocity equation with respect to time. For the exercise, this results in the equation \( \alpha = 8.0 + 12.0t \).
  • At \( t = 2 \) seconds, plugging into this equation, we find the angular acceleration to be \( 32.0 \) radians per second squared.
  • Since the equation includes terms dependent on time \( t \), the angular acceleration is not constant. As time varies, so does the angular acceleration.
Kinematics
Kinematics, in the context of rotational motion, involves studying the motion without reference to the forces causing it. It includes concepts such as angular position, angular velocity, and angular acceleration.
The equations governing these quantities help in predicting the motion of rotating objects. In rotational kinematics, these quantities can be derived from each other through differentiation or integration.
  • Angular position tells the rotational orientation and can be monitored using time-dependent equations like \( \theta = 2.0 + 4.0t^2 + 2.0t^3 \).
  • Angular velocity is obtained by differentiating the angular position, leading to equations like \( 8.0t + 6.0t^2 \).
  • Angular acceleration is derived from angular velocity, providing a complete picture of an object's rotation dynamics through \( \alpha = 8.0 + 12.0t \).
These concepts work together to describe the complete motion profile of an object in rotational motion.

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Most popular questions from this chapter

At 7: 14 A.M. on June 30,1908 , a huge explosion occurred above remote central Siberia, at latitude \(61^{\circ} \mathrm{N}\) and longitude \(102^{\circ}\) \(\mathrm{E} ;\) the fireball thus created was the brightest flash seen by anyone before nuclear weapons. The Tunguska Event, which according to one chance witness "covered an enormous part of the sky," was probably the explosion of a stony asteroid about \(140 \mathrm{~m}\) wide. (a) Considering only Earth's rotation, determine how much later the asteroid would have had to arrive to put the explosion above Helsinki at longitude \(25^{\circ} \mathrm{E}\). This would have obliterated the city. (b) If the asteroid had, instead, been a metallic asteroid, it could have reached Earth's surface. How much later would such an asteroid have had to arrive to put the impact in the Atlantic Ocean at longitude \(20^{\circ} \mathrm{W} ?\) (The resulting tsunamis would have wiped out coastal civilization on both sides of the Atlantic.)

At \(t=0\), a flywheel has an angular velocity of \(4.7 \mathrm{rad} / \mathrm{s}, \mathrm{a}\) constant angular acceleration of \(-0.25 \mathrm{rad} / \mathrm{s}^{2}\), and a reference line at \(\theta_{0}=0 .\) (a) Through what maximum angle \(\theta_{\max }\) will the reference line turn in the positive direction? What are the (b) first and (c) second times the reference line will be at \(\theta=\frac{1}{2} \theta_{\max } ?\) At what (d) negative time and (e) positive time will the reference line be at \(\theta=10.5\) rad? (f) Graph \(\theta\) versus \(t\), and indicate the answers to (a) through (e) on the graph.

Calculate the rotational inertia of a wheel that has a kinetic energy of \(24400 \mathrm{~J}\) when rotating at \(602 \mathrm{rev} / \mathrm{min}\).

Four particles, each of mass, \(0.20 \mathrm{~kg}\), are placed at the vertices of a square with sides of length \(0.50 \mathrm{~m}\). The particles are connected by rods of negligible mass. This rigid body can rotate in a vertical plane about a horizontal axis \(A\) that passes through one of the particles. The body is released from rest with rod \(A B\) horizontal (Fig. \(10-61\) ). (a) What is the rotational inertia of the body about axis \(A ?\) (b) What is the angular speed of the body about axis \(A\) when rod \(A B\) swings through the vertical position?

Between 1911 and 1990 , the top of the leaning bell tower at Pisa, Italy, moved toward the south at an average rate of \(1.2 \mathrm{~mm} / \mathrm{y}\). The tower is \(55 \mathrm{~m}\) tall. In radians per second, what is the average angular speed of the tower's top about its base?
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