A hollow, spherical shell with mass 2.00 kg rolls without slipping down a 38.0^∘ slope. (a) Fin

A hollow, spherical shell with mass 2.00 kg rolls without...

Key Concepts

Dynamics on an Inclined Plane

Analyzing motion on an inclined plane involves resolving forces into components parallel and perpendicular to the surface. The gravitational force component parallel to the incline causes the object to accelerate, while the perpendicular component is balanced by the normal force. This framework is essential for applying Newton's laws to objects rolling down slopes.

Dependence of Acceleration on Mass

In systems where rolling without slipping occurs, the acceleration can become independent of the mass of the object because both the translational and rotational equations of motion include the mass, leading to a cancellation. This principle illustrates that doubling the mass of the object does not necessarily alter the acceleration or frictional requirements.

Static Friction and Its Role in Rolling

Static friction is the force that resists the initiation of sliding between two surfaces in contact. In rolling motion, static friction provides the necessary torque to prevent slipping and ensure that the rolling without slipping condition is maintained. Its maximum value is determined by the coefficient of static friction times the normal force.

Moment of Inertia

The moment of inertia is a measure of an object’s resistance to changes in its rotational motion around an axis. For objects with mass distributed at a distance from the axis of rotation, such as a hollow sphere, the moment of inertia plays a key role in determining the angular acceleration for a given torque.

Rolling Without Slipping

This condition occurs when an object rolls in such a way that its point of contact with the surface does not slide. It establishes a kinematic relationship between the translational velocity (or acceleration) of the center of mass and the angular velocity (or acceleration), ensuring that the linear and rotational motions are synchronized.

Newton's Second Law for Rotation

This law describes the rotational analog of Newton's second law, where the net torque acting on an object is equal to its moment of inertia multiplied by its angular acceleration. It helps in understanding how friction at the contact point provides the torque necessary for rotational acceleration.

Newton's Second Law for Translation

This principle explains the relationship between the net force acting on an object and the resulting acceleration, stating that the force equals the mass times the acceleration. It is used to analyze the motion of objects moving along an incline by decomposing the gravitational force into components parallel and perpendicular to the incline.

Transcript

00:01 In this problem, we're looking at a ball that is rolling without slipping down an incline of beta, where beta is equal to 38 degrees.

00:09 The sphere is a spherical shell with a mass of two kilograms.

00:15 And given those two pieces of information, we're supposed to figure out what the force of friction is, what its acceleration is, and what the coefficient of friction working on it is.

00:31 So we're going to start out with we're going to start out just finding equations for those things we're going to start with what we what we know about the system which is ft is equal to ma the total force is going to be equal to m a and then it will be equal to the downhill force fd minus the force of friction.

00:56 I'm just pulling it uphill which in turn will be equal to m g sign beta and we know that the force of friction is equal to coefficient of friction times normal force we know the normal force is equal to mg cosine beta so this will be equal to minus coefficient of friction times mg cosine beta so next we should solve for here we have another we have another equation that will bring in the friction force and acceleration and that will be the torque equation we have we know that torque is going to be equal to for this the only thing applying a torque is the friction force since it's the only thing that is acting on the edge of the sphere rather than the whole sphere itself so that will be equal to friction force times r.

02:10 And we also know that the torque is equal to the moment of inertia times the angular acceleration.

02:19 The moment of inertia of spherical shell is two -thirds m r squared and the angular acceleration is equal to the acceleration over r.

02:36 And we know that this acceleration is going to be the same because, of course, it's rolling without slipping.

02:41 So however fast the ball is moving, the bottom of the ball has to be moving the same speed to avoid slipping.

02:49 So this will be equal to these two things multiplied by each other.

02:53 So two thirds, mr squared times a over r.

03:00 These will cancel.

03:02 And then this will cancel with this over there.

03:05 You'll be left with f is equal to two -thirds ma.

03:15 We know that that is equal to this.

03:17 Maybe i shouldn't substitute it out immediately, but i'm just going to substitute it back in.

03:24 Ma is equal to m -g -sign beta minus r new f, two -thirds ma.

03:38 Right away, i'll just cancel out all these ms...

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