Rinse and trim the strawberries, then puree. Peel and quarter the apple. Remove the core, and finely grate. Weigh the fruit, and adjust the amount of sugar if necessary. The sugar should weigh half of what the fruit does.
Mix the fruit in a saucepan with the jam sugar. Bring to a boil, stirring occasionally. Add the pepper and cook for 4 minutes. Stir in the balsamic vinegar, and stir to combine. Fill the jam into the hot prepared jars and seal tightly.