Amulet Holy Symbol

Amulet

An Amulet is a Holy Symbol that is bejeweled or painted to channel divine magic. A Cleric or Paladin can use this item as a Spellcasting Focus. Amulets must be worn or held.

Material (M)

A Material component is a particular material used in a spell’s casting, as specified in parentheses in the Components entry. These materials aren’t consumed by the spell unless the spell’s description states otherwise. The spellcaster must have a hand free to access them, but it can be the same hand used to perform Somatic components, if any.

Spellcasting Focus

A Spellcasting Focus is an object that certain creatures can use in place of a spell’s Material components if those materials aren’t consumed by the spell and don’t have a cost specified. Some classes allow its members to use certain types of Spellcasting Focuses.

To use a Spellcasting Focus, you must hold it unless its description says otherwise (see “Equipment” for descriptions).

A character which is holding a spellcasting focus in one hand and something else in the other can cast spells with BOTH somatic and material components without a free hand.

This allows a paladin or a cleric to cast SM spells without unequipping their Emblem Shield or their weapon for instance.

I was wondering what happen when you use an amulet holy symbol that can simply be worn to be used (holding is not required). Must you have a free hand to cast the SM spell in this case?

You always need a free hand, save for the exceptions you noted. It’s a balance thing with keeping hands occupied. In this case, you might flavor it like, they need to touch the amulet or something. 
What the holy symbol is is really just flavor text, the same rules apply to all of them.

Quote from Jayce_Danathar >>

Must you have a free hand to cast the SM spell in this case?

You never have to "touch" an amulet.

The way that you get there is by following this sequence of rules:

A Material component is a particular material used in a spell’s casting, as specified in parentheses in the Components entry . . . The spellcaster must have a hand free to access them.

So, if for some reason the Cleric does not use any spellcasting focus and doesn't have a Component Pouch, then they must collect and carry around the actual material components on their person in order to be able to cast the corresponding spell.  In those cases, the Cleric must physically "access" these materials with a free hand.

However, instead of doing that, a Cleric might use a Spellcasting Focus instead:

If a spell doesn’t consume its materials and doesn’t specify a cost for them . . . the spellcaster can substitute a Spellcasting Focus if the caster has a feature that allows that substitution.

. . .

to use a Spellcasting Focus, you must hold it unless its description says otherwise.

Cleric Feature -- Level 1:  Spellcasting

Spellcasting Focus. You can use a Holy Symbol as a Spellcasting Focus for your Cleric spells.

So, a Cleric can substitute a Holy Symbol as a Spellcasting Focus for the actual material components.  With that substitution, how does the Cleric use the Holy Symbol when casting a spell?  By default (as above), he must hold it unless its description says otherwise.  So, what does its description say about this?  As it turns out, it depends on which Holy Symbol is being used:

A Holy Symbol takes one of the forms in the Holy Symbol table and is bejeweled or painted to channel divine magic. A Cleric or Paladin can use a Holy Symbol as a Spellcasting Focus.

The table indicates whether a Holy Symbol needs to be held, worn, or borne on fabric (such as a tabard or banner) or a Shield.

From the Table:

Amulet (worn or held)

1 lb.

5 GP

Or, from the description of the Amulet:

Amulets must be worn or held

An amulet can be used as a Spellcasting Focus by simply wearing it.  By using it in this manner, a free hand is not required when casting an M spell.

However, as written there is technically a problem when it comes to casting an "SM" spell with this setup.  It has to do with the requirement for the S component.  This is very likely a design error or implementation error by the creators of the game.  We know this because there have been official Sage Advice style written examples from the game designers where it has been stated that a Cleric is required to have a free hand when casting an "S" (but no "M") spell but is fine with a sword-and-board setup to cast an "SM" spell.  Unfortunately, as written, that doesn't actually work because of how these rules are written:

A Somatic component is a forceful gesticulation or an intricate set of gestures. A spellcaster must use at least one of their hands to perform these movements.

and:

The spellcaster must have a hand free to access them, but it can be the same hand used to perform Somatic components, if any.

It was probably intended that not needing a free hand to access the material component would also meet the free hand requirement for the S component.  However, when examining those two rules closely, that conclusion does not logically follow.  You actually always need a free hand to perform the S component.  You just do not need another free hand for the M component.

Although this is an unfortunate oversight, there is still some utility to using a worn amulet that other types of spellcasters do not enjoy.  If you are casting a spell that does have an "M" component but does NOT have an "S" component, then you don't need any free hand while using a worn amulet.  Useful for spells such as Light and Tongues but not much else . . .

It's probably a reasonable house rule to allow a worn amulet to satisfy the free hand requirement of both "M" and "SM" spells but not "S" spells (as apparently intended) since there is a decent number of "S" spells and also a decent number of "SM" spells.