Osteoarthritis (to some degree) is an inevitable part of the aging process.
I think that limiting physical activity for kids is more likely to lead to childhood (and possibly, adult) obesity. Obesity is a contributing factor to the development of osteoarthritis, not to mention a variety of other significant health problems.
Most children want and need to be active. Physical activity is essential for motor skill acquisition, bone and joint modelling/development, neurosensory development, to name just a few.
Getting out and active with kids is a wonderful part of parenting - one of my great pleasures is cycling with my 8 year old son. I think that your daughter will limit her own activity when she is tired.
Paul