Book edition 7th

Author(s) Douglas C. Giancoli

Pages 978 pages

ISBN 978-0321625922

Answers without the blur.

Just sign up for free and you're in.

Short Answer

A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.20 m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket.
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?

(a) The speed of the bucket at the bottom of the circle is $1.797 m / s$ .

(b) The speed of the bucket at the top of the circle is $3.43 m / s$ . The bucket will move faster when the rope does not slack.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Understanding the application of Newton’s second law in the bucket motion

A bucket is tied with a rope, and it executes circular motion. From Newton’s second law, the net force acting in the vertical direction will be equated to the force acting on the bucket. This force is in the force of centripetal force.

By using these equations, the speed at the top or bottom of the circle can be evaluated.

Step 2. Identification of given data

The given data can be listed below as,

The radius of circle is, $r = 1.20 m$ .
The mass of the bucket is, $m = 2 kg$ .
The tension in the rope is, $T = 25 N$ .
The acceleration due to gravity is, $g = 9.81 m / s^{2}$ .

Step 3. Representation of the forces on the bucket at the bottom of the circle

The forces on the bucket can be shown as,

Here, mg is the weight of the bucket, $F_{T}$ is the tension force in the rope, m is the mass of the bucket.

Step 4. (a) Determination of the speed of the bucket at the bottom of the circle

From the Newton’s second law, the equation can be expressed as,

$\begin{matrix} \sum F_{y} = F_{c} \\ F_{T} - m g = m a_{c} \\ \frac{m v^{2}}{r} = F_{T} - m g \\ v = \sqrt{\frac{r (F_{T} - m g)}{m}} \end{matrix}$

Here, v is the bucket’s speed, $F_{c}$ is the centripetal force, $a_{c}$ is the centripetal acceleration of the bucket.

Substitute the values in the above equation.

$\begin{matrix} v = \sqrt{\frac{1.20 m (25 N (\frac{1 kg \cdot m / s^{2}}{1 N}) - 2 kg \times 9.81 m / s^{2})}{2 kg}} \\ v = \sqrt{3.228} m / s \\ v = 1.797 m / s \end{matrix}$

Thus, the speed of the bucket at the bottom of the circle is $1.797 m / s$ .

Step 5. Representation of the forces on the bucket at the top of the circle

The forces on the bucket can be shown as,

From the Newton’s second law, the equation can be expressed as,

$\begin{matrix} \sum F_{y} = F_{c} \\ F_{T} + m g = m a_{c} \\ \frac{m v^{2}}{r} = F_{T} + m g \\ v = \sqrt{\frac{r (F_{T} + m g)}{m}} \end{matrix}$

Step 6. (b) Determination of the speed of the bucket at the top of the circle

The rope will not go to slack. So, the tension in the rope will become zero.

Now, the above equation can be written as,

$\begin{matrix} v = \sqrt{\frac{r (0 + m g)}{m}} \\ = \sqrt{\frac{r m g}{m}} \\ = \sqrt{r g} \end{matrix}$

Substitute the values in the above equation.

$\begin{matrix} v = \sqrt{1.20 m \times (9.81 m / s^{2})} \\ v = \sqrt{11.772} m / s \\ v = 3.43 m / s \end{matrix}$

Thus, the speed of the bucket at the top of the circle is $3.43 m / s$ .

From the above two cases, (a) and (b), it can be concluded that the bucket will move faster at the top of the circle when the rope does not slack.

Find Study Materials for

Create Study Materials

Select your language

Physics Principles with Applications

Answers without the blur.

Short Answer

A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.20 m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket.
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?

Step by Step Solution

Step 1. Understanding the application of Newton’s second law in the bucket motion

Step 2. Identification of given data

Step 3. Representation of the forces on the bucket at the bottom of the circle

Step 4. (a) Determination of the speed of the bucket at the bottom of the circle

Step 5. Representation of the forces on the bucket at the top of the circle

Step 6. (b) Determination of the speed of the bucket at the top of the circle

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Select your language

Physics Principles with Applications

Answers without the blur.

Short Answer

A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.20 m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N.(a) Find the speed of the bucket.(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?

Step by Step Solution

Step 1. Understanding the application of Newton’s second law in the bucket motion

Step 2. Identification of given data

Step 3. Representation of the forces on the bucket at the bottom of the circle

Step 4. (a) Determination of the speed of the bucket at the bottom of the circle

Step 5. Representation of the forces on the bucket at the top of the circle

Step 6. (b) Determination of the speed of the bucket at the top of the circle

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Work Energy and Power

Radiation

Astrophysics

Physics of Motion

Scientific Method Physics

Famous Physicists

Fluids

Torque and Rotational Motion

Nuclear Physics

Fields in Physics

Measurements

Space Physics

Electricity

Physical Quantities and Units

Medical Physics

Electrostatics

Further Mechanics and Thermal Physics

Mechanics and Materials

Engineering Physics

Energy Physics

Force

Particle Model of Matter

Waves Physics

Magnetism

Modern Physics

Atoms and Radioactivity

Dynamics

Electricity and Magnetism

Conservation of Energy and Momentum

Fundamentals of Physics

Kinematics Physics

Circular Motion and Gravitation

Linear Momentum

Rotational Dynamics

Oscillations

Translational Dynamics

Geometrical and Physical Optics

Thermodynamics

Magnetism and Electromagnetic Induction

Electromagnetism

Electric Charge Field and Potential

Turning Points in Physics

A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.20 m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket.
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?