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Chapter 8: Problem 108

The angular momentum vector for a rotating object is given by the following: $$ \vec{L}=3 t \hat{x}+4 t^{2} \hat{y}+0.5 t^{3} \hat{z} $$ Calculate the torque as a function of time associated with the rotational motion (about the same axis). What is the magnitude of the torque at \(t=2 \mathrm{~s}\) ?

Short Answer

Expert verified
The torque as a function of time is \( \vec{韦} = 3\hat{x} + 8t \hat{y} +1.5t^{2}\hat{z}\). The magnitude of the torque at \(t = 2\) seconds is \( \sqrt{301\,}\, N.m\).

Step by step solution

01

Writing Down the Given

Write down the given angular momentum vector \(\vec{L}=3t\hat{x}+ 4t^{2}\hat{y}+0.5t^{3}\hat{z}\)
02

Determine the Torque Vector

Since torque \(\vec{韦}\) is the time derivative of the angular momentum \(\vec{L}\), we find the derivative of \(\vec{L}\) with respect to time to get the torque vector. \(\vec{韦} = d\vec{L}/dt\). Differentiate each component of the angular momentum vector with respect to time.
03

Calculate the Torque Vector

When differentiating, we get \(\vec{韦} = 3\hat{x}+ 8t\hat{y}+1.5t^{2}\hat{z}\). So, torque as a function of time is \(\vec{韦} = 3\hat{x}+ 8t\hat{y}+1.5t^{2}\hat{z}\).
04

Calculate the Magnitude of Torque

To find the torque at \(t=2\) seconds, substitute \(t\) with \(2\) in the torque equation. Then compute its magnitude (length of the vector), i.e., square root of the sum of the squares of its components. The magnitude of a vector \(\vec{v} = v_{1}\hat{x}+ v_{2}\hat{y}+v_{3}\hat{z}\) is given by \(\sqrt{{v_{1}^{2} + v_{2}^{2} + v_{3}^{2}}}\).
05

Final Calculation

Substituting \(t = 2\) seconds into \(\vec{韦}\), we get \(\vec{韦} = 3\hat{x} + 16\hat{y} + 6\hat{z}\). Using the formula for the magnitude of the vector, the magnitude of \(\vec{韦}\) is \(\sqrt{{9 + 256 + 36}} = \sqrt{301\,}\, N.m\).

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Key Concepts

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

Understanding Angular Momentum
Angular momentum is a fundamental concept in rotational motion. It describes how much "spin" an object has. Imagine a figure skater spinning on ice. Their speed and the distribution of their mass determine the angular momentum. In many ways, it's the rotational equivalent of linear momentum.

Angular momentum for a rotating object can be represented as a vector, like \[\vec{L} = 3t\hat{x} + 4t^2\hat{y} + 0.5t^3\hat{z}\]This equation tells us how the angular momentum changes over time along different axes:
  • The \( t \) variable indicates time.
  • Each component shows how angular momentum changes in the \( \hat{x}, \hat{y}, \) and \( \hat{z} \) directions.
The significance of angular momentum lies in its conservation in a closed system, providing insights into the behavior of spinning bodies.
Time Derivative in Physics
The time derivative is an essential tool in physics for understanding how quantities change over time. When analyzing motion, scientists often need to see how variables evolve.

In the context of rotational motion, the time derivative of angular momentum gives us the torque. Mathematically, it鈥檚 denoted as:
  • \(\vec{韦} = \frac{d\vec{L}}{dt}\)
This formula reveals the rate of change of angular momentum with respect to time. In simple terms, torque measures how much force is applied to twist or rotate an object. Solving the derivative of the given angular momentum vector gives the torque vector as a function of time, indicating the influence causing the change in rotation.
Exploring Vector Magnitude
Vector magnitude refers to the 'size' or 'length' of a vector. Vectors can point in various directions, each having its own magnitude.

To compute the magnitude of a vector like \(\vec{韦} = 3\hat{x} + 16\hat{y} + 6\hat{z}\), use the formula:
\[||\vec{v}|| = \sqrt{v_1^2 + v_2^2 + v_3^2}\]Plugging in our values, it becomes:
  • \(\sqrt{3^2 + 16^2 + 6^2}\)
  • Resulting in \(\sqrt{301}\)
Magnitude helps determine how intense a vector is without concern for its direction. In the context of torque, it tells us the strength of the rotational force at a given time.
The Dynamics of Rotational Motion
Rotational motion is all around us, from spinning wheels to the Earth's rotation. It involves objects rotating around an axis and is governed by principles similar to linear motion but adapted for rotation.

Key components of rotational motion include:
  • **Angular Displacement**: How far an object rotates, akin to distance in linear motion.
  • **Angular Velocity**: The rate of change of angular displacement, similar to linear velocity.
  • **Angular Acceleration**: The change in angular velocity over time.
Understanding rotational motion requires considering how forces like torque affect an object's spin. It provides insight into how objects gain speed or come to a stop when rotating, vital for engineering and physics applications.

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Most popular questions from this chapter

What is the speed of an electron in the lowest energy orbital of hydrogen, of radius equal to \(5.29 \times 10^{-11} \mathrm{~m}\) ? The mass of an electron is \(9.11 \times 10^{-31} \mathrm{~kg}\) and its angular momentum in this orbital is \(1.055 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\).

In 1932 Albert Dremel of Racine, Wisconsin, created his rotary tool that has come to be known as a dremel. (a) Suppose a dremel starts from rest and achieves an operating speed of \(35,000 \mathrm{rev} / \mathrm{min}\). If it requires \(1.2 \mathrm{~s}\) for the tool to reach operating speed and it is held at that speed for \(45 \mathrm{~s}\), how many rotations has the bit made? Suppose it requires another \(8.5 \mathrm{~s}\) for the tool to return to rest. (b) What are the angular accelerations for the start-up and the slowdown periods? (c) How many rotations does the tool complete from start to finish?

Calculate the final speed of a uniform, solid sphere of radius \(5 \mathrm{~cm}\) and mass \(3 \mathrm{~kg}\) that starts with a translational speed of \(2 \mathrm{~m} / \mathrm{s}\) at the top of an inclined plane that is \(2 \mathrm{~m}\) long and tilted at an angle of \(25^{\circ}\) with the horizontal. Assume the sphere rolls without slipping down the ramp.

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A spinning top completes 6000 rotations before it starts to topple over. The average speed of the rotations is \(800 \mathrm{rpm}\). Calculate how long the top spins before it begins to topple.
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