Problem 46 - University Physics with Modern Physics

Question

A light, flexible rope is wrapped several times around a hollow cylinder, with a weight of 40.0 \(\mathrm{N}\) and a radius of 0.25 \(\mathrm{m}\) , that rotates without friction about a fixed horizontal axis. The cylinder is attached to the axle by spokes of a negligible moment of inertia. The cylinder is initially at rest. The free end of the rope is pulled with a constant force \(P\) for a distance of \(5.00 \mathrm{m},\) at which point the end of the rope is moving at 6.00 \(\mathrm{m} / \mathrm{s}\) . If the rope does not slip on the cylinder, what is the value of \(P ?\)

Step-by-Step Solution

Answer

The force \( P \) is 14.688 N.

Step 1: Understand the Problem

We need to determine the force \( P \) applied to a rope that causes a hollow cylinder to rotate without slipping. The cylinder has a weight of 40.0 N and a radius of 0.25 m. The rope is pulled 5.00 m, and the end velocity of the rope as it is pulled is 6.00 m/s. This is a problem related to rotational motion and dynamics.

Step 2: Identify Known Quantities

Weight of the cylinder \( W = 40.0 \) N, radius \( r = 0.25 \) m, distance pulled \( d = 5.00 \) m, final velocity \( v = 6.00 \) m/s. Based on the weight, the mass \( m \) of the cylinder can be calculated using gravitational acceleration \( g = 9.8 \text{ m/s}^2 \).

Step 3: Calculate the Mass of the Cylinder

The mass \( m \) of the cylinder is calculated as:\[ m = \frac{W}{g} = \frac{40.0 \text{ N}}{9.8 \text{ m/s}^2} = 4.08 \text{ kg}\]

Step 4: Calculate the Moment of Inertia

For a hollow cylinder, the moment of inertia \( I \) is calculated using the formula:\[ I = m r^2\]Substituting the known values:\[ I = 4.08 \text{ kg} \cdot (0.25 \text{ m})^2 = 0.255 \text{ kg} \cdot \text{m}^2\]

Step 5: Apply Work-Energy Principle

According to the work-energy principle, the work done by the force \( P \) should equal the change in kinetic energy:\[ W = \Delta KE = \frac{1}{2} I \omega^2\]where \( \omega = \frac{v}{r} = \frac{6.00 \text{ m/s}}{0.25 \text{ m}} = 24.0 \text{ rad/s} \).

Step 6: Calculate Rotational Kinetic Energy

Insert \( \omega \) into the equation for kinetic energy:\[ \Delta KE = \frac{1}{2} \cdot 0.255 \text{ kg} \cdot \text{m}^2 \cdot (24.0 \text{ rad/s})^2 = 73.44 \text{ J}\]

Step 7: Calculate the Force \( P \) Using Work Formula

Work \( W \) is also expressed as:\[ W = P \cdot d\]Thus,\[ 73.44 \text{ J} = P \cdot 5.00 \text{ m}\]Solving for \( P \):\[ P = \frac{73.44 \text{ J}}{5.00 \text{ m}} = 14.688 \text{ N}\]

Step 8: Ensure Validity

Review calculations and ensure each assumption is based on physics principles. Check each step links logically and mathematically to the next.

Key Concepts

Moment of Inertia

The moment of inertia is a crucial concept in understanding rotational dynamics. It refers to the measure of an object's resistance to change in its rotation. Think of it as the rotational equivalent of mass in linear motion.
For a hollow cylinder, such as the one in our problem, the moment of inertia can be calculated using the formula:

• \[ I = m r^2 \]

where \( m \) is the mass of the cylinder and \( r \) is its radius.
The above calculation shows that a hollow cylinder has its mass concentrated further from the axis compared to a solid cylinder, which gives it a higher moment of inertia for the same mass and radius.

• It's crucial to understand that the distribution of mass affects how easily the cylinder can be rotated.

The higher the moment of inertia, the harder it is to start or stop the rotation.

Work-Energy Principle

The work-energy principle is fundamental in analyzing systems involving motion. It tells us that the work done by all forces acting on an object is equal to the change in its kinetic energy. This relationship is essential in understanding how forces interact with movement in different forms, such as rotational motion in our problem.
In this scenario, the work-energy principle states that the work \((W)\), done by the force \((P)\) in pulling the rope:

• \[ W = P imes d \]

is equal to the change in kinetic energy of the rotating cylinder:

• \[ rac{1}{2} I imes \omega^2 \]

where \( I \) is the moment of inertia and \( \omega \) is the final angular velocity.
Thus, understanding the work done by the force is crucial for calculating how it contributes to the motion of rotating systems._section_break_rejection_warning_ The energy initially present and the energy added through work are converted into rotational kinetic energy.

Rotational Motion

Rotational motion refers to the motion of an object around a fixed axis. It's analogous to linear motion, but instead of moving in a straight line, the object rotates.
In our exercise, the cylinder rotates around its central axis, which is fixed, and we consider how this rotation is initiated and sustained by the pulling force \((P)\).
Several key aspects define rotational motion:

• Angular displacement: How far in degrees or radians an object rotates.
• Angular velocity: The rate of rotation around the axis.

Since friction is negligible, it plays no part in reducing velocity over time, allowing the calculation to focus solely on the energy aspects of motion.
This makes it easier to examine changes without worrying about external frictional forces. Understanding rotational motion helps to simplify many problems where circular paths are involved, providing insight into how forces translate into rotational movement.

Angular Velocity

Angular velocity is a vector quantity that represents how fast an object rotates. It’s measured in radians per second and is crucial in understanding how quickly parts of a rotating system are moving over time.
In our problem, the angular velocity \((\omega)\) was calculated as:

• \[ \omega = \frac{v}{r} \]

where \( v \) is the linear velocity of the rope and \( r \) is the radius of the cylinder.
Angular velocity is important because:

• It helps us determine how fast an object is spinning.
• It’s a crucial component in the calculation of rotational kinetic energy.

Connecting this with other concepts like the moment of inertia and work-energy principle helps in fully understanding the rotational effects of pulling the rope around the cylinder. By controlling angular velocity, applying forces, and visualizing its impact on objects, one can predict motion patterns effectively.