A mass m = 11 kg rests on a frictionless table and accelerated by a spring with

Question

A mass m = 11 kg rests on a frictionless table and accelerated by a spring with spring constant k = 4938 N/m. The floor is frictionless except for a rough patch. For this rough path, the coefficient of friction is mu k = 0.45. The mass leaves the spring at a speed v = 4.1 m/s. How much work done by the spring as it accelerates the mass? How far was the spring stretched from its unstreched length? The mass is measured to leave the rough spot with a final speed V_f = 2.1 m/s. How much work is done by friction as the mass crosses the rough spot? What is the length of the rough spot? In a new scenario, the block only makes it (exactly) half-wa

Explanation / Answer

1) work done by spring = 0.5 * 11 * 4.1 * 4.1

                                       = 92.45 J

2)   string stretched length = sqrt(11 * 4.1 * 4.1/4938)

                                              = 0.1935 m

3) work done by friction = 92.45 - (0.5 *11*2.1 * 2.1)

                                       = 68.195 J

4) length of rough spot = 68.195/(0.45 * 11*9.8)

                                     = 1.4058 m

5)   Here, work done by friction = 0.45 * 11 * 9.8 * 0.7029

                                                    = 34.0976 J

   =>   spring was compressed = sqrt(2 * 34.0976/4938)

                                                  = 0.1175 m

6)   here, coefficient of friction = (0.5 * 4938 * 0.1175 * 0.1175)/(11 * 9.8 * 1.4058)

                                                    = 0.2249

7) magnitude of work done by friction is three times greater .

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