Question

A uniform solid cylinder rolls without slipping down an incline. A hole is drilled through the cylinder along its axis. The radius of the hole is 0.50 times the (outer) radius of the cylinder. (a) Does the cylinder take more or less time to roll down the incline now that the hole has been drilled? Explain. (b) By what percentage does drilling the hole change the time for the cylinder to roll down the incline? (W) tutorial: rolling)

Short answer

The cylinder takes less time to roll down (approximately 13.4% less).

Step-by-step solution

Understand the System

We have a solid cylinder with a hole through its center rolling down an incline. The radius of the hole is half of the outer radius of the cylinder. We need to compare the time it takes for this cylinder to roll down an incline with the original solid cylinder.

Assess the Moment of Inertia

The moment of inertia for a solid cylinder of outer radius \( R \) is \( I = \frac{1}{2}mR^2 \). For the cylinder with a hole, the new moment of inertia is increased because mass is removed. So, \( I' = \frac{1}{2}mR^2 - \frac{1}{2}m(0.5R)^2 = \frac{1}{2}mR^2(1 - \frac{1}{4}) = \frac{3}{8}mR^2 \). Since moment of inertia is reduced, its angular acceleration will be increased, causing it to roll faster.

Compare Angular Dynamics

Angular acceleration \( \alpha \) is related to torque \( \tau \) and moment of inertia \( I \) by the equation \( \tau = I \alpha \). With a smaller inertia, the same torque will cause a larger angular acceleration for the hollow cylinder, implying that it speeds up faster.

Analyze Effect on Rolling Time

Faster angular acceleration implies that it takes less time for the cylinder to reach the bottom of the incline. This means the cylinder with the hole takes less time to roll down compared to the solid cylinder.

Calculate Time Ratio

If the solid cylinder's time to roll down is \( t \), and the hollow one is \( t' \), then \( t'/t = \sqrt{\frac{I'}{I}} = \sqrt{\frac{3/8mR^2}{1/2mR^2}} = \sqrt{\frac{3}{4}} \). Thus, \( t' = t \times \sqrt{\frac{3}{4}} \).

Calculate Percentage Change

Percentage change in time is calculated by \( \left(\frac{t - t'}{t}\right) \times 100\% = \left(1 - \sqrt{\frac{3}{4}}\right) \times 100\% \approx 13.4\% \). The cylinder takes approximately 13.4% less time to roll down the incline.

Key concepts

Rolling Motion

When dealing with rolling motion, we are referring to the type of movement that occurs when an object rolls along a surface. In the case of a uniform solid cylinder, it rolls down an incline without slipping. Rolling without slipping means that every point of contact on the cylinder surface touches the incline without skidding. This implies a direct relationship between linear velocity and angular velocity. Essentially, the distance the cylinder covers linearly as it rolls is synchronized with its rotational movement. This concept is crucial because it allows us to analyze the energy distribution between translational and rotational motion. For cylinders, or any rolling objects, maintaining a balance between these energies is vital to understanding how they will behave as they roll. When a hole is drilled into the cylinder, it alters this balance slightly, affecting the moment of inertia and rolling dynamics.

Angular Acceleration

Angular acceleration is a measure of how quickly an object speeds up its rotation. It’s an essential aspect when evaluating the rolling motion of a cylinder or any rotating body. The formula linking angular acceleration to torque and moment of inertia is:\[\tau = I \alpha\]Where:

• \(\tau\) is the torque applied,
• \(I\) is the moment of inertia,
• \(\alpha\) is the angular acceleration.

In our exercise, the cylinder with a hole has a reduced moment of inertia compared to a solid one. This means any given torque will cause a greater angular acceleration, effectively increasing how fast it rotates. As a result, the cylinder with a hole reaches the bottom of the incline in less time than a solid cylinder, because it speeds up its rotation more efficiently.

Torque

Torque is essentially a rotational force that causes an object to rotate around an axis. It’s like a twist that can accelerate the spinning of an object. Torque’s role in our exercise is pivotal because it’s the force that propels the cylinder down the incline. The relationship between torque and rolling objects is part of what determines their movement. A cylinder that experiences torque accelerates its rotational speed, depending on its moment of inertia. With the drilled cylinder showing less inertia, any given torque will be more effective in rolling motion, enhancing both angular and linear acceleration. This effectiveness of torque on reduced inertia explains why changes in structure (like drilling a hole) significantly affect how fast and smoothly a cylinder will roll down an incline.

Incline Dynamics

Incline dynamics involve the interaction of gravitational forces, friction, and the geometry of rolling objects, as they travel down a slope. An object like a cylinder experiences gravitational pull down the slope causing it to accelerate. The angle of the incline, the friction between the surfaces, and the distribution of mass within the object all come into play. In our exercise, drilling a hole modifies how mass is distributed, leading to changes in moment of inertia and, therefore, in dynamics of acceleration. By changing the mass distribution, and consequently the moment of inertia, the descent speed changes. A cylinder will reach the bottom faster because its moment of inertia is lower. This ties into how efficiently the cylinder can convert potential energy into kinetic energy during its descent, further showcasing the importance of understanding incline dynamics in rolling motion scenarios.