How have federal mandates allowed the federal government to control the traditional police powers of the state?

Money, money, money. The federal government provides monetary help to states for various matters normally within a state's own jurisdiction. But, this help always comes with conditions or mandates that the states must meet or they don't get the money. For example, a basic state police power is setting speed limits on local roads. The federal government has no authority over local state highways, because they are not involved in interstate commerce. Say the federal government believes a speed limit should not exceed 60 miles per hour, but the state believes 65 is just fine. The federal government can offer financial assistance to states for maintenance of local roads but mandate that in order to receive that money, the speed limit cannot exceed 60 miles perhour. States that need this money are now forced to comply with federal guidelines rather than its own, even though the federal government has no power to set speed limits on local highways. This happens in many other areas as well. Education, housing, health care and many others. If a state does not go along with a federal mandate, no money.