With regard to the dentist himself, it doesn't. Strict liability is the imposition of legal responsibility without regard to a defendant’s reasonable care or fault. [Goodwin v. Reilley (1985) 176 Cal.App.3d 86, 221 Cal.Rptr. 374 (drunk driving was not ultrahazardous activity for strict liability purposes)] Strict liability requires a finding that a "defect" exists.
The defect can take one of three forms:
(1) a design defect;
(2) a flaw in the manufacturing or building process, so that the result differs from what was intended; or
(3) the absence of a warning that is necessary to allow the product’s safe use. [Maloney v. American Pharmaceutical Co. (1988) 207 Cal.App.3d 282, 255 Cal.Rptr. 1 (under facts of case successor drug manufacturer was not liable for predecessor’s product)]
If you wish to use any tort other than "negligence", besides suing the dentist for his negligent failure to inspect, maintain and repair his property, you'll have to also sue the manufacturer of the chair.
Be aware, however, that strict liability under these circumstances has a Statute of Limitations; that is, if the chair is, for example, more than 10 years old, it is highly doubtful that you'll be able to succeed against the manufacturer.