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Chapter 11: Problem 5

Rotational Position The rotational position of a point on a rotating wheel is given by \(\theta=2.0 \mathrm{rad}+\left(4.0 \mathrm{rad} / \mathrm{s}^{2}\right) t^{2}+\left(2.0 \mathrm{rad} / \mathrm{s}^{3}\right) t^{3}\), where \(\theta\) is in radians and \(t\) is in seconds. At \(t_{1}=0\), what are (a) the point's rotational position and (b) its rotational velocity? (c) What is its rotational velocity at \(t_{3}=4.0 \mathrm{~s} ?\) (d) Calculate its rotational acceleration at \(t_{2}=2.0 \mathrm{~s}\). (e) Is its rotational acceleration constant?

Short Answer

Expert verified
(a) 2.0 rad, (b) 0 rad/s, (c) 128 rad/s, (d) 32 rad/s虏, (e) No

Step by step solution

01

Calculate rotational position at t=0

Substitute t=0 into the equation for \( \theta \): \[ \theta = 2.0 + (4.0)t^2 + (2.0)t^3 \] So, \[ \theta(0) = 2.0\] Therefore, at t = 0, the rotational position is 2.0 rad.
02

Calculate rotational velocity function

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

Calculate rotational velocity at t=0

Substitute t=0 into the equation for \( \omega \) : \[ \omega = 8.0t + 6.0t^2 \] So, \[ \omega(0) = 8.0(0) + 6.0(0)^2 = 0 \] Therefore, at t = 0, the rotational velocity is 0 rad/s.
04

Calculate rotational velocity at t=4s

Substitute t=4 into the equation for \( \omega \) : \[ \omega = 8.0(4) + 6.0(4)^2 = 32 + 96 = 128 \] Therefore, at t = 4s, the rotational velocity is 128 rad/s.
05

Calculate rotational acceleration function

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

Calculate rotational acceleration at t=2s

Substitute t=2 into the equation for \( \alpha \) : \[ \alpha = 8.0 + 12.0(2) = 8.0 + 24 = 32 \] Therefore, at t = 2s, the rotational acceleration is 32 rad/s虏.
07

Check if rotational acceleration is constant

The equation for rotational acceleration is \[ \alpha = 8.0 + 12.0t \]. Since it depends on \ t \, the rotational acceleration is not constant.

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Key Concepts

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

rotational position
Rotational position refers to the location of a point on a rotating object, described by an angle (胃) in radians. In this example, we have the equation \( \theta = 2.0 \, \text{rad} + (4.0 \, \text{rad/s}^2) \, t^2 + (2.0 \, \text{rad/s}^3) \, t^3 \). For any given time (t), we plug in the value of t to find the angle. This tells us how far a point has rotated relative to a reference point.

For instance, when t = 0, substituting in the equation, we get \( \theta(0) = 2.0 \). Thus, at t = 0, the point鈥檚 rotational position is 2.0 radians.
rotational velocity
Rotational velocity \( (\text{蠅}) \) is the rate at which the rotational position changes. It is the first derivative of the rotational position with respect to time. Given \( \theta = 2.0 + 4.0t^2 + 2.0t^3 \), to find the rotational velocity, we differentiate:

\( \text{蠅} = \frac{d\theta}{dt} = 8.0t + 6.0t^2 \)

By plugging in any time value (t), we can find the instantaneous rotational velocity at that moment. For t = 0, the rotational velocity is 0 rad/s. For t = 4s, substituting gives \( \text{蠅}(4) = 128 \) rad/s, showing how quickly the position changes over time.
rotational acceleration
Rotational acceleration \( (\text{伪}) \) is the rate at which the rotational velocity changes. It is found by taking the first derivative of the rotational velocity with respect to time. Given \( \text{蠅} = 8.0t + 6.0t^2 \), we differentiate to find:

\( \text{伪} = \frac{d\text{蠅}}{dt} = 8.0 + 12.0t \)

To find the rotational acceleration at specific times, we substitute the time (t) into this equation. For example, at t = 2s, \( \text{伪}(2) = 32 \) rad/s虏. Importantly, since \( \text{伪} \) depends on t, it is not constant鈥攊t changes with time.
derivatives in physics
Derivatives in physics are powerful tools for understanding how quantities change. They help us transition from one physical concept to another. For instance:
  • Rotational position (胃) gives us the location.
  • The derivative of 胃 with respect to time gives us rotational velocity (蠅), describing how fast the position changes.
  • The derivative of 蠅 with respect to time gives us rotational acceleration (伪), explaining how fast velocity changes.

This chain from 胃 to 蠅 to 伪 showcases how rates of change (derivatives) provide deeper insights into physical motion.

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Most popular questions from this chapter

A Pulley A pulley, with a rotational inertia of \(1.0 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}\) about its axle and a radius of \(10 \mathrm{~cm}\), is acted on by a force applied tangentially at its rim. The force magnitude varies in time as \(F=\) \((0.50 \mathrm{~N} / \mathrm{s}) t+\left(0.30 \mathrm{~N} / \mathrm{s}^{2}\right) t^{2}\), with \(F\) in newtons and \(t\) in seconds. The pulley is initially at rest. At \(t=3.0 \mathrm{~s}\) what are (a) its rotational acceleration and (b) its rotational speed?

Turntable Two A record turntable is rotating at \(33 \frac{1}{3}\) rev/min. A watermelon seed is on the turntable \(6.0 \mathrm{~cm}\) from the axis of rotation. (a) Calculate the translational acceleration of the seed, assuming that it does not slip. (b) What is the minimum value of the coefficient of static friction, \(\mu^{\text {stat }}\), between the seed and the turntable if the seed is not to slip? (c) Suppose that the turntable achieves its rotational speed by starting from rest and undergoing a constant rotational acceleration for \(0.25 \mathrm{~s}\). Calculate the minimum \(\mu^{\text {stat }}\) required for the seed not to slip during the acceleration period.

A Diver A diver makes \(2.5\) revolutions on the way from a \(10-\mathrm{m}\) high platform to the water. Assuming zero initial vertical velocity, find the diver's average rotational velocity during a dive.

Two Solid Cylinders Two uniform solid cylinders, each rotating about its central (longitudinal) axis, have the same mass of \(1.25 \mathrm{~kg}\) and rotate with the same rotational speed of \(235 \mathrm{rad} / \mathrm{s}\), but they differ in radius. What is the rotational kinetic energy of (a) the smaller cylinder, of radius \(0.25 \mathrm{~m}\), and \((\mathrm{b})\) the larger cylinder, of radius \(0.75 \mathrm{~m}\) ?

Small Ball A small ball of mass \(0.75 \mathrm{~kg}\) is attached to one end of a \(1.25\) -m-long massless rod, and the other end of the rod is hung from a pivot. When the resulting pendulum is \(30^{\circ}\) from the vertical, what is the magnitude of the torque about the pivot?
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