A solid sphere is released from height h from the top of an incline making an angle θwith the ho

A solid sphere is released from height h from the top of an...

Key Concepts

Dynamics on an Incline and Time of Descent

The motion along an inclined plane involves resolving gravitational forces into components parallel and perpendicular to the incline. The acceleration along the incline, which affects the time taken to descend, differs between rolling without slipping and frictionless sliding. The difference in acceleration affects the time intervals required for an object to reach the bottom of the incline in each case.

Frictionless Sliding

In the case of frictionless sliding, the object moves without rotating, which means that all of the gravitational potential energy is converted solely into translational kinetic energy. This simplifies the analysis and typically results in a different final speed compared to the rolling case because none of the energy is diverted into rotation.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. In analyses of rolling motion, knowing the moment of inertia is essential for calculating the rotational kinetic energy when an object is rolling without slipping.

Translational and Rotational Kinetic Energy

When an object is rolling without slipping, its kinetic energy is the sum of translational kinetic energy (associated with its center of mass motion) and rotational kinetic energy (associated with its rotation about the center of mass). This partitioning is important in determining the final speed because the energy is divided between these two forms, unlike in the case of frictionless sliding where only translational kinetic energy is present.

Conservation of Energy

This principle states that the total mechanical energy of a system remains constant if only conservative forces are acting. In the context of objects on an incline, gravitational potential energy is converted into kinetic energy, which may include translational and rotational components. This concept is used to relate the initial height of an object to its speed at the bottom of the incline.

Rolling Without Slipping

This concept refers to a type of motion where an object rolls in such a way that its point of contact with the surface is momentarily at rest relative to that surface. This condition establishes a kinematic relationship between the linear velocity and the angular velocity of the object, which is crucial for correctly applying energy conservation to problems involving rolling objects.

Transcript

00:01 In this exercise, we have a sphere that is on the top of an inclined plane of height h, and that makes an angle of theta with the horizontal.

00:13 And the sphere is released from rest, and in question a, we have to find what is the final speed of the sphere if the sphere rolls without slipping.

00:29 So basically what rolling without sleeping means is that the sphere will rotate with an angular speed omega that is equal to v, that is the linear speed of the sphere divided by the radius r.

00:55 And basically, what we need to do, first of all, is to notice that the moment of inertia i of the sphere is equal to two -fifths of m are squared, and that the kinetic energy that's associated with the rotation of the sphere is equal to i.

01:28 Times omega squared divided by 2.

01:34 And now in order to calculate the final speed, v, of the sphere, i'm going to use conservation of energy.

01:45 So basically, remember the conservation of energy means that the initial potential energy, ui, plus the initial kinetic energy, k -i, is equal to the final potential energy, uf, plus the final potential, the final kinetic energy, kf.

02:03 Now since the sphere starts at rest, ki is zero, and i'm going to choose the zero of the potential energy to be at the bottom of the inclined plane.

02:19 So this means that the final potential energy is zero.

02:24 So basically what we have is that the initial potential energy is equal to the final kinetic energy.

02:32 Now the initial potential energy is just m, sorry, it's m, g .h, while the final kinetic energy is equal to the rotational kinetic energy, that's i omega squared over two, plus the mass, the translational kinetic energy, which is the mass v squared over two.

02:58 So this is i, v squared, over 2r plus m v squared over 2 now i as we said before and i is 2 5th of m r squared so here actually here we should have an r squared of course so we have this is equal to two 5th v squared divided by r squared plus m v squared over 2 and this is equal to actually there's a 2 here so this is equal to 1 5th of m v squared plus 1 half of mv squared and this is equal to 7 mv squared over 10 and now you're left with m gh is equal to 7tenths of mc tenths of m mv squared.

04:11 So the ms cancel out and v is equal to the square root of 10 g8 dh over 7...

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