Question

A solid sphere and a solid cylinder having same mass and radius roll down the same incline the ratio of their acceleration will be.... \(\\{\mathrm{A}\\} 15: 14\) \(\\{\mathrm{B}\\} 14: 15\) \(\\{\mathrm{C}\\} 5: 3\) \(\\{\mathrm{D}\\} 3: 5\)

Short answer

The ratio of accelerations between the solid sphere and the solid cylinder is \( \frac{a_{sphere}}{a_{cylinder}} = \frac{20}{21} \). However, the given options do not match this result, indicating possible errors or discrepancies in the answer options.

Step-by-step solution

Define the problem parameters and forces

Let us first define the parameters for the problem: - m: mass of the sphere and the cylinder (mass is equal for both) - R: radius of the sphere and the cylinder (radius is equal for both) - θ: angle of the incline - g: acceleration due to gravity Both sphere and cylinder experience gravitational force (mg) and frictional force.

Determine Moment of Inertia

We need to find the moment of inertia for both the sphere and the cylinder to analyze their motion and acceleration. For a solid sphere: Moment of Inertia (I_sphere) = \( \frac{2}{5}mR^2 \) For a solid cylinder: Moment of Inertia (I_cylinder) = \( \frac{1}{2}mR^2 \)

Determine Acceleration equations

Using the conservation of energy and moment of inertia, we can derive the equations for the sphere and the cylinder's accelerations down the incline. For the solid sphere: Torque = I_sphere * Angular_Acceleration_sphere mg * R * sin(θ) = \( \frac{2}{5}mR^2 \) * \( \frac{a_{sphere}}{R} \) Solving for a_sphere: \( a_{sphere} = \frac{5}{7}g\sin(\theta) \) For the solid cylinder: Torque = I_cylinder * Angular_Acceleration_cylinder mg * R * sin(θ) = \( \frac{1}{2}mR^2 \) * \( \frac{a_{cylinder}}{R} \) Solving for a_cylinder: \( a_{cylinder} = \frac{3}{4}g\sin(\theta) \)

Determine the ratio of the accelerations

Now, we need to find the ratio of the accelerations of the sphere and the cylinder. \( \frac{a_{sphere}}{a_{cylinder}} = \frac{\frac{5}{7}g\sin(\theta)}{\frac{3}{4}g\sin(\theta)} \) Cancel out g and sin(θ) from both the numerator and the denominator: \( \frac{a_{sphere}}{a_{cylinder}} = \frac{\frac{5}{7}}{\frac{3}{4}} = \frac{5}{7} \times \frac{4}{3} = \frac{5 \times 4}{3 \times 7} = \frac{20}{21} \) So, the ratio of the accelerations of solid sphere to solid cylinder is \( 20:21 \). However, the given options do not match the result. This indicates that the question might have some errors or discrepancies in the answer options. It is advisable to consult the textbook or teacher for clarification.

Key concepts

Moment of Inertia

When it comes to understanding the motion of rolling objects, the moment of inertia is crucial. This concept is similar to mass but for rotational motion. It represents how much torque is needed for a certain angular acceleration around an axis.

Think of it as the rotational equivalent of mass in linear motion. Each rotating body, depending on its shape and axis of rotation, has a unique formula for calculating its moment of inertia.

• For a solid sphere, the moment of inertia, denoted as \(I_{sphere}\), is \( \frac{2}{5}mR^2 \).
• For a solid cylinder, the moment of inertia \(I_{cylinder}\) is \( \frac{1}{2}mR^2 \).

The differences in these formulas arise from how mass is distributed in each shape. It’s the "spread" of mass that affects how much effort is required to set the object into rotational motion. A sphere with mass distributed farther from its axis will have a lower moment of inertia compared to a cylinder with mass closer to its axis. This relationship impacts their accelerations when rolling down an incline.

Conservation of Energy

One of the fundamental principles in physics is the conservation of energy. It states that energy cannot be created or destroyed, only transformed from one form to another. In the context of rolling motion, this principle is valuable for analyzing how objects move without slipping.

When a sphere or a cylinder rolls down an incline, potential energy (due to height) is converted into both translational and rotational kinetic energy. Here’s how energy transformation occurs:

• Potential energy at the top: \( mgh \) (where \(h\) is the height of the incline).
• At the bottom, this energy is converted into kinetic: \( \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \).

This equation accounts for both linear velocity \(v\) and angular velocity \(\omega\). The moment of inertia \(I\) influences how the total energy is divided between rotational and translational motion. This allows us to determine velocities and accelerations, leading to a clear understanding of the motion dynamics of the sphere and cylinder.

Torque and Angular Acceleration

Torque and angular acceleration are vital in explaining how objects start rotating. Torque is a rotational force that causes an object to spin, similar to how a push or pull causes an object to slide. It’s calculated by multiplying the force applied by the radius at which it acts times the sine of the angle involved.

In rolling motion, the force causing the torque is gravity, acting along the incline creating torque which results in angular acceleration. This relationship is shown as:

• Torque = Moment of Inertia \( \times \) Angular Acceleration

For both the sphere and the cylinder:

• The gravitational force \(mg\sin(\theta)\) acts at a distance \(R\) from the axis causing these bodies to rotate.
• The angular acceleration \(\alpha\) is the change in angular velocity per time unit and directly linked to the net torque applied.

The different moments of inertia for the sphere and the cylinder lead to different expressions for their angular accelerations. By carefully examining these expressions, we can calculate how quickly each object will roll down the incline, highlighting the fascinating relationship between linear and angular dynamics in rolling motion.