A smooth sphere of radius R is made to translate in a straight line with a constant acceleration

step 1 In this question, we have a smooth sphere of radius R. It's made to translate with a constant ac

A smooth sphere of radius R is made to translate in a...

Key Concepts

Non?Inertial Reference Frames

When analyzing motion as seen from an accelerating sphere, it is necessary to treat the sphere’s frame as non?inertial. In such frames, additional apparent or fictitious forces appear, which are not present in inertial frames. These fictitious forces must be introduced to correctly account for the deviations from Newton's laws as normally formulated, influencing the motion of objects observed within that frame.

Fictitious (Pseudo) Forces

In a non?inertial frame, fictitious forces (or pseudo-forces) arise due to the acceleration of the reference frame itself. In this problem, the constant acceleration of the sphere gives rise to an inertial force acting on the particle opposite to the acceleration. This force modifies the effective forces acting on the particle and must be included in any energy or force analysis.

Energy Conservation in Non-Inertial Systems

Even in non?inertial frames, energy conservation can be applied if one accounts for the work done by both real and fictitious forces. Here, the energy approach involves equating the change in kinetic energy of the particle relative to the sphere with the work done by gravitational and pseudo forces as the particle slides along the curved surface.

Constrained Motion on a Curved Surface

The motion of the particle is restricted to the surface of a sphere, which is a classic example of constrained motion. This constraint means that the particle's motion must be described using appropriate coordinates (such as the angular coordinate ? along the surface) and that the normal reaction force does no work. The analysis typically involves decomposing the forces along tangential and radial directions of the sphere.

Generalized Coordinates

Using an angular coordinate such as ? to describe the position of the particle on a spherical surface simplifies the analysis by reducing the problem to one involving a single variable. This approach is especially useful in deriving relationships between the speed of the particle and its position, via energy methods or differential equations that incorporate the geometry of the constraint.

Transcript

00:01 In this question, we have a smooth sphere of radius r.

00:04 It's made to translate with a constant acceleration, a is equal to g.

00:08 The particle is kept initially at the top of the sphere and released from there with a velocity equal to zero with respect to the sphere.

00:15 We want to find the speed of the particle with respect to the sphere as a function of the angle theta as it slides down.

00:23 So in this diagram, we can see the particle at the top of the sphere, and we can see the particle as it has slid down.

00:29 We know that the force acts on the sphere towards the left, so the particle experiences a pseudo -force equal to m .a.

00:36 Towards the right.

00:37 The initial kinetic energy of the particle is equal to zero, because it starts with zero velocity.

00:43 As a result, using the work energy theorem, we can say that the initial gravitational potential energy is equal to the final kinetic energy, if we take the final position to have zero gravitational potential energy.

00:55 We can also work out the change in height, thus the change in gravitation, potential energy by resolving the components here as seen so that we can say that the change in height so h is equal to r1 minus cosine theta.

01:15 We also know that this distance between here is equal to r sine theta as it's the opposite component to this angle.

01:24 So what we can say now is that the work done by gravity is equal to wg and the work done by the pseudo force is equal to wf...

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