M3 - Circular motion question

There's something I'm very confused about, when you have a particle moving on the top of the outer surface of a sphere. Why is it that the centripetal force is equal to the normal reaction plus the component of the weight acting towards the centre? I'd've thought that the weight would be 'helping' the centripetal force, and so the reaction would equal the weight plus the centripetal force. Why is this not the case?

your question is typical of a student which does maths and physics.

the centripetal force is a "fictitious force" which only exists in the physics vocabulary.
the are only two forces acting on the particle, assuming smooth.

the weight
the normal reaction

Remember that the centripetal force as TeeEm said is a 'fictitious force' - it is the RESULTANT force towards the centre of a circle.
Well if at the top and on the outside of a sphere, taking inwards as positive, mg-R= centripetal force.

That seems very fair. But in the mark scheme in my above post, they add the reaction to the weight instead of subtracting it. What's going on?

Yes, R is negative as they've defined inwards as positive but in any case, it really doesn't matter as R=0. Perhaps the (+R) is to indicate the inclusion of a reaction force, rather than an addition.

Ah, that's fine then. Thanks for the help!

There's something I'm very confused about, when you have a particle moving on the top of the outer surface of a sphere. Why is it that the centripetal force is equal to the normal reaction plus the component of the weight acting towards the centre? I'd've thought that the weight would be 'helping' the centripetal force, and so the reaction would equal the weight plus the centripetal force. Why is this not the case?

your question is typical of a student which does maths and physics.

the centripetal force is a "fictitious force" which only exists in the physics vocabulary.
the are only two forces acting on the particle, assuming smooth.

the weight
the normal reaction

(..)

I think that it's at best "non standard" to call the centripetal force "fictitious" and at worst simply wrong.

The centripetal force ("centre seeking force") is the force required to keep an object moving on a curved path. It's a real force generated by real interactions, such as by the pull of a string, or the reaction at a physical surface.

Usually a force is called fictitious if it has to be invoked to explain motion that is seen by a non-inertial observer, such as one who is accelerating linearly or rotating.

So, for example, suppose you see a mass travelling at constant speed about a fixed point, due its being tied to a string. Then a non-rotating observer sees that the velocity vector of the mass is changing, and hence by N2 he realises that the vector sum of the forces acting on it is non-zero; this corresponds with the fact that he can see that the string produces the required force.

On the other hand, an observer rotating in the same direction and at the same rate as the mass sees the mass as stationary. However, he also sees a string, which must be pulling it toward the centre. By N2, he realises that since it is stationary, there must be another force acting on the mass of equal size to that produced by the string, but in the opposite direction.

He is aware that he and the mass are rotating, so he therefore says that when a mass rotates it experiences two forces, one of which pulls it away from the centre; he calls this the centrifugal force ("centre fleeing force") but it's fictitious since it doesn't arise due to any real interaction or object, and he only has to invoke it due to the fact that he is in a rotating frame.

Fictitious forces are only needed to make Newton's laws work from the point of view of an accelerating observer, and the centripetal force in this example doesn't fall into that category.