Problem 1
A block slides from rest down a frictionless track through vertical height , then enters a vertical loop of radius . Find the speed at the loop bottom. Assume ideal objects and components, neglect unstated losses, and use SI units. (a) Derive the primary requested quantity from the appropriate law. (b) Analyze quantitatively how the result depends on the given parameters. (c) Check a physically meaningful limiting case and explain its meaning. State sign conventions, show reasoning, and check dimensions. (a) Calculate the requested quantity from the data. (b) Give the law or expression showing parameter dependence. (c) Examine a limit consistent with the model. The vertical loop has radius R; angle theta is measured from the upward vertical at its center, and the block slides on the inner surface.
problems.proof.stepOf
problems.proof.analysis
Rearrange algebraically and select the solution consistent with initial conditions and physical meaning. Check dimensions before interpreting the expression.
problems.proof.pitfall. Keep the sign convention and constraints consistent; do not replace a signed quantity by its magnitude without justification.