Problem 1

A particle of mass

m

is attached to a light string of length

L

and moves in a vertical circle. Its speed at the lowest point is

v_0

. Find its speed at the highest point and the string tension at both points; assume the string remains taut. (a) State the assumptions and establish the governing equation for the requested quantity. (b) Solve it in terms of the given symbols, identifying the direction, sign, or physical nature of the result. (c) Check a simple limiting case and state when the model remains valid. Use SI units consistently, retain the symbols in the setup, and enforce all geometric constraints and initial conditions.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

T_t+mg=\frac{mv_t^2}{L}\Rightarrow T_t=m(\frac{v_0^2}{L}-5g)

problems.proof.analysis

At the top both weight and tension point inward. Substitution gives the result; tautness requires $v_0^2\ge5gL$ .

problems.proof.pitfall. Do not discard an initial condition or stated constraint; an algebraic root outside the physical domain is not an answer.