Question

A thin ring, a solid sphere, a hollow spherical shell, and a disk of uniform thickness are placed side by side on a wide ramp of length \(\ell\) and inclined at angle \(\theta\) to the horizontal. At time \(t=0\), all four objects are released and roll without slipping on parallel paths down the ramp to the bottom. Friction and air resistance are negligible. Determine the order of finish of the race.

Short answer

Based on the comparison of accelerations, the order in which the objects will finish rolling down the inclined ramp is: 1. Solid Sphere 2. Uniform Disk 3. Hollow Sphere (Spherical Shell) 4. Thin Ring

Step-by-step solution

Determine the moment of inertia for each object.

For a uniform disk, hollow sphere, solid sphere, and thin ring, the moment of inertia is given by: Uniform Disk: \(I_\text{disk} = \frac{1}{2} MR^2\) Hollow Sphere (Spherical Shell): \(I_\text{hollow} = \frac{2}{3} MR^2\) Solid Sphere: \(I_\text{solid} = \frac{2}{5} MR^2\) Thin Ring: \(I_\text{ring} = MR^2\) where \(M\) is the mass and \(R\) is the radius.

Use the conservation of energy to find the final velocity of each object.

Since air resistance and friction are negligible, we can use the conservation of energy principle: Potential Energy(at top of the ramp) = Kinetic Energy(at bottom of the ramp) \(Mgh = \frac{1}{2} mv^2 + \frac{1}{2} Iv^2/R^2\) where \(h\) is the vertical height of the ramp, \(v\) is the object's final linear velocity, and \(I\) is the moment of inertia. Cancel the mass (M) on both sides, and solve for the final velocity (v): \(v = \sqrt{\frac{2gh}{1 + I/R^2}}\)

Determine the acceleration for each object.

We can use the no-slip condition, which states that the linear acceleration (\(a\)) and angular acceleration (\(\alpha\)) are related: \(a = R \alpha\) Replace \(\alpha\) with \(Ia/R^2\) (from the equation \(I\alpha = MR^2 \alpha\)): \(a = \frac{IR}{MR^2}\) Now, we can use the final velocity (v) and the distance (l) traveled by each object, and the following kinematic equation: \(v^2 = 2al\) Solve for the acceleration (a): \(a = \frac{v^2}{2l}\)

Compare the accelerations and determine the order of finish.

From step 3, we have the expression for acceleration: \(a = \frac{gy}{1 + I/R^2}\) The object with the highest acceleration will reach the bottom of the ramp first. We can now substitute the moment of inertia for each object in the equation and compare the accelerations: \(a_\text{disk} = \frac{gy}{1+\frac{1}{2}} = \frac{2}{3}gy\) \(a_\text{hollow} = \frac{gy}{1+\frac{2}{3}} = \frac{3}{5}gy\) \(a_\text{solid} = \frac{gy}{1+\frac{2}{5}} = \frac{5}{7}gy\) \(a_\text{ring} = \frac{gy}{1+1} = \frac{1}{2}gy\) Comparing the accelerations, we can determine the order of finish: 1. Solid Sphere 2. Uniform Disk 3. Hollow Sphere (Spherical Shell) 4. Thin Ring

Key concepts

Moment of Inertia

Understanding moment of inertia is crucial in the realm of physics, especially when analyzing rolling motion data. It's a measure of an object's resistance to changes in its rotation. Think of it like the rotational equivalent of mass for linear motion. The moment of inertia depends not only on the mass of an object but also on how that mass is distributed relative to the axis of rotation.

For objects rolling down an incline, such as our given exercise, each shape has a unique moment of inertia. A solid sphere has less inertia resisting its motion compared to a hollow sphere, disk, or ring because its mass is closer to the axis of rotation, represented in the formulas provided in the solution. Therefore, it can accelerate more quickly down the ramp. This underlines the importance of not only the mass but the distribution of mass in dictating how an object will move when it's set into rotation.

Conservation of Energy

The principle of conservation of energy plays a fundamental role in various physical scenarios, including the motion of objects down a ramp. It tells us that the total energy in a closed system remains constant over time. In the context of our exercise, this principle allows us to equate the initial potential energy at the top of the ramp with the sum of translational and rotational kinetic energy at the bottom.

Using the conservation of energy, we can derive the final velocity for each rolling object as shown in the solution. This is because the gravitational potential energy lost as the object rolls down is converted into kinetic energy. Since there's no external force like friction or air resistance at play, the energy transition is straightforward, demonstrating how conservation of energy can be an elegant way to understand motion without delving into the specifics of forces involved.

Kinematic Equations

The kinematic equations are the tools that let us describe the motion of objects with respect to time without the need to understand the forces involved. These equations are a cornerstone in physics because they link the displacement, velocity, acceleration, and time of a moving object.

In the exercise, the kinematic relationship between final velocity squared and twice the product of acceleration and distance (\(v^2 = 2al\)) is used after determining acceleration from the earlier steps. The use of these equations in conjunction with the concept of rolling without slipping aids in determining the time or velocity at any point of the objects' journey down the ramp. When appropriately applied, they reveal that the solid sphere, with its highest acceleration, will finish the race down the incline first, followed by the disk, hollow sphere, and thin ring, respectively. This showcases how combining kinematic equations with other physical principles can powerfully predict the outcome of dynamic systems.