Question

A solid sphere rolls without slipping down an incline, starting from rest. At the same time, a box starts from rest at the same altitude and slides down the same incline, with negligible friction. Which object arrives at the bottom first? a) The solid sphere arrives first. b) The box arrives first. c) Both arrive at the same time. d) It is impossible to determine.

Short answer

Answer: b) The box arrives first.

Step-by-step solution

Analyze the forces acting on the objects

In order to calculate the acceleration of both the solid sphere and the box, we need to analyze the forces acting on them. The force acting along the incline for both objects is the gravitational force component along the incline, mg sinθ, where m is the mass and g is the acceleration due to gravity. For a solid sphere, we also need to account for the rolling motion, which is determined by a rotational inertia. The moment of inertia (rotational inertia) for a solid sphere is given by the formula: I = 2/5 * m * R², where R is the radius of the sphere.

Calculate the acceleration of the box

The box slides down the incline, so we consider only the linear motion. Newton's second law states that the force acting on an object is equal to the mass times acceleration (F = ma). For the box, the force acting on it is equal to mg sinθ: mg sinθ = ma a = g sinθ The acceleration of the box along the incline is g sinθ.

Calculate the acceleration of the solid sphere

For the solid sphere, we need to consider both the linear and rotational motion. Using Newton's second law, the force acting on the sphere along the incline is also equal to mg sinθ: mg sinθ = ma + τ where τ is the torque produced by the force acting on the sphere. The torque is equal to the moment of inertia times the angular acceleration (τ = I * α), and α is related to the linear acceleration by a = α * R (since it is rolling without slipping). From the equation above, we have: mg sinθ = ma + (2/5 * m * R²)(a/R) Simplifying the equation, we get: a = (5/7)g sinθ

Compare the accelerations

Now that we have calculated the acceleration of both the box and the solid sphere, we can compare them: a_box = g sinθ a_sphere = (5/7)g sinθ Since g sinθ > (5/7)g sinθ, the box has a greater linear acceleration than the solid sphere.

Determine which object reaches the bottom first

Since the box has a greater linear acceleration than the solid sphere, it will reach the bottom of the incline first. Therefore, the correct answer is: b) The box arrives first.

Key concepts

Rolling Without Slipping

When a solid sphere rolls down an incline, it maintains a condition known as 'rolling without slipping'. This means the point of the sphere in contact with the incline does not slide; instead, it has a momentary velocity of zero relative to the incline. To achieve this, the rotational motion of the sphere, characterized by angular velocity \( \omega \), is intricately linked to its linear motion, represented by the linear velocity \( v \). The relationship between these two is expressed by the equation \( v = \omega \times R \), where \( R \), is the radius of the sphere.

This concept ensures direct contact with the surface at all times, and because energy is not lost to kinetic friction, it results in a combination of rotational and linear kinetic energies contributing to the sphere's total kinetic energy. Understanding this will aid students in visualizing why the sphere begins its motion with a lower linear acceleration compared to an object that is slipping down the slope.

Moment of Inertia

The moment of inertia, often denoted by \( I \), is a quantity expressing an object's tendency to resist angular acceleration; this is the rotational equivalent to mass in linear motion. Each object has a unique moment of inertia that depends on its mass distribution relative to the axis of rotation.

For our solid sphere rolling without slipping down the incline, the moment of inertia is crucial in determining how it accelerates. The formula \( I = \frac{2}{5} mR^{2} \) represents the moment of inertia for a solid sphere about its center. This tells us that a larger sphere or one with more mass will generally have a higher moment of inertia, implying a greater resistance to changes in its rotational state. When students grasp the concept of moment of inertia, they'll better understand the dynamics of rotational motion and why distribution of mass affects an object's acceleration.

Newton's Second Law

Newton's second law is foundational for understanding motion: \( F = ma \)—force equals mass times acceleration. This law applies to rotational motion, with a slight twist: the torque \( \tau \) is analogous to force, the moment of inertia \( I \) corresponds to mass, and angular acceleration \( \alpha \) is akin to linear acceleration.

The tweaked version of Newton's second law for rotation is \( \tau = I\alpha \). By comprehending this principle, students will be able to calculate how forces affect motion, both rotational and linear. It's especially helpful in problems like our solid sphere on an incline, as it helps to establish the relationship between forces, motion, and energy in both linear and rotational contexts.

Gravitational Force Component

Objects on an incline experience a component of gravitational force pulling them down the slope. This force is less than the full force of gravity, proportional to the incline's angle. It's calculated as \( mg\sin(\theta) \), where \( m \) is the object's mass, \( g \) is the acceleration due to gravity, and \( \theta \) is the incline's angle.

Understanding this gravitational force component is vital in predicting how objects behave on slopes. For the box in our problem, the force is the product of the object's mass and the gravitational field; the only motion is linear, so this component governs its acceleration down the slope. Educators should emphasize this concept to help students identify the different forces at play when objects are placed on an inclined plane.

Linear and Rotational Motion

Linear motion is the straightforward movement from one point to another without any rotating. Rotational motion, on the other hand, involves an object turning around an axis. The beauty of physics lies in how these concepts coalesce, such as in the scenario of the solid sphere rolling down an incline.

When the sphere rolls without slipping, its center of mass undergoes linear motion while also rotating about its center, displaying rotational motion. This dual nature of motion requires an understanding of both linear and angular acceleration, velocity, and displacement. For instance, in our problem, the sphere has a lower linear acceleration due to its energy being divided between translating down the slope and rotating about its center. By focusing on these concepts together, students can develop a holistic perspective of the mechanics that describe the movement of objects around us.