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Magazine Chapter 7: Problem 96
A hoop rolls on a horizontal ground without slipping with linear speed \(v\). Speed of a particle \(P\) on the circumference of the hoop at angle \(\theta\) is (a) \(2 v \sin \frac{\theta}{2}\) (b) \(v \sin \theta\) (c) \(2 v \cos \frac{\theta}{2}\) (d) \(v \cos \theta\)
Short Answer
Expert verified
The correct answer is (a) \(2 v \sin \frac{\theta}{2}\).
Step by step solution
01
Understand the Motion of the Hoop
When a hoop rolls on the ground without slipping, every point on the hoop has both translational and rotational motion. The center of the hoop moves with a linear speed \(v\).
02
Analyze the Velocity in Terms of the Hoop's Motion
A particle on the circumference of the hoop moves with a combination of the translational velocity of the center and the rotational velocity around the center. The total velocity vector of the particle is determined by combining these two effects.
03
Velocity Components of Particle P
The velocity of point \(P\) at angle \(\theta\) can be broken down into components: one due to the translational motion of the center (which is \(v\)) and another due to the rotational motion of the hoop (which is also \(v\)). The rotational velocity at point \(P\) can be split into components: \(v \cos \theta\) in the direction opposite to translation, and \(v \sin \theta\) perpendicular to it.
04
Combine Velocity Components
The horizontal (backward along the circumference) component of velocity at \(P\) opposes the translational component, reducing it to \(v - v \cos \theta\). The perpendicularly upward component is \(v \sin \theta\).
05
Total Speed Calculation Using Trigonometry
Using the Pythagorean theorem, the magnitude of the total velocity of particle \(P\) is calculated as: \[\sqrt{(v - v \cos \theta)^2 + (v \sin \theta)^2} = 2v \sin \frac{\theta}{2}.\]
06
Select the Correct Answer
From the derived expression, the correct option for the speed of the particle \(P\) at angle \(\theta\) is (a) \(2 v \sin \frac{\theta}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Kinematics
Kinematics is the branch of physics that studies the motion of objects without considering the forces that cause the motion. In our exercise, a hoop is rolling without slipping along a horizontal plane. The hoop's overall motion is described by kinematic principles.
The hoop's center moves with a linear speed \(v\), which quantifies how fast the hoop is shifting position. For a particle located on the hoop's circumference, we must consider both this linear motion and its rotation to understand its movement fully.
The hoop's center moves with a linear speed \(v\), which quantifies how fast the hoop is shifting position. For a particle located on the hoop's circumference, we must consider both this linear motion and its rotation to understand its movement fully.
- Linear Motion: Relates to the straight-line movement of the hoop's center.
- Rotational Motion: Describes the circular movement of the particles on the hoop around its center.
Rotational Motion
In addition to moving in a straight line, the hoop undergoes rotational motion. This is where each point on the hoop spins around the center. The hoop's rotation is synchronized with its linear movement in a specific way, characterized by rolling without slipping.
This means that points on the hoop's edge don't slide against the ground, ensuring that the distance covered by the hoop's center each second equals the hoop's circumference traveled.
This means that points on the hoop's edge don't slide against the ground, ensuring that the distance covered by the hoop's center each second equals the hoop's circumference traveled.
- Angular Velocity: The rate at which the hoop rotates. Each point on the circumference covers a similar angular distance in a given time.
- Combine Rotational and Linear Motion: The outer points have additional velocity due to the hoop spinning.
Trigonometry
In this exercise, trigonometry is crucial for breaking down and understanding the different components of velocity for a point on the hoop's circumference. Trigonometry helps analyze how the rotational components contribute to the total motion of a particle at a given angle \(\theta\).
- Angle \(\theta\): Used to determine the position of point \(P\) on the hoop.
- Sine and Cosine Functions: Help to break the rotational velocity into perpendicular components. For example, \(v \cos \theta\) (opposite to translational direction) and \(v \sin \theta\) (perpendicular to it).
- Combining Components: Calculating the resultant velocity involves summing these components using trigonometric identities and geometric methods, like the Pythagorean theorem.
Velocity Components
Velocity components give us a structured way to analyze the movement of the particle \(P\) on the hoop. The particle's movement results from two primary influences: translational motion of the entire hoop and the hoop's rotation. By determining how these velocities combine, we can find the particle's actual speed.
- Translational Component: A constant velocity of \(v\) due to the linear motion of the hoop's center.
- Rotational Components: Consist of two parts 鈥� one aligned with the direction of motion, \(v \cos \theta\), and another perpendicular to it, \(v \sin \theta\).
- Total Velocity: Calculated by vector addition, considering both components, leading to the formula \(\sqrt{(v - v \cos \theta)^2 + (v \sin \theta)^2}\).
