In each of the questions below, hanging a mass m from a spring stretches the spring l meters from its unloaded length. calculate the spring constant k, write the differential equation that governs the motion of the undamped mass-spring system, and find the solution that satisfies the initial conditions specified. units are mks; gravitational constant is 9.8 m/s2.

Zeros that follow non-zeros numbers and are also to the right of a decimal point are significant

The frequency of vibrations of a vibrating violin string is given by f = 1 2l t ρ where l is the length of the string, t is its tension, and ρ is its linear density.† (a) find the rate of change of the frequency with respect to the following. (i) the length (when t and ρ are constant) (ii) the tension (when l and ρ are constant) (iii) the linear density (when l and t are constant) (b) the pitch of a note (how high or low the note sounds) is determined by the frequency f. (the higher the frequency, the higher the pitch.) use the signs of the derivatives in part (a) to determine what happens to the pitch of a note for the following. (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates df dl 0 and l is ⇒ f is ⇒ (ii) when the tension is increased by turning a tuning peg df dt 0 and t is ⇒ f is ⇒ (iii) when the linear density is increased by switching to another string df dρ 0 and ρ is ⇒ f is ⇒

Select for each of the following statements whether it is correct or incorrect. (a) in an isothermal expansion of an ideal gas. (b) the temperature remains constant. (b) the pressure remains constant. (c) there is work done by the gas. (d) there is heat added to the gas. (e) the change in internal energy equals zero.