A spool whose inner core has a radius of 1.00 cm and whose end caps have a radiu

Question

A spool whose inner core has a radius of 1.00 cm and whose end caps have a radius of 1.50 cm has a string tightly wound around the inner core. The spool is free to roll without slipping on a horizontal surface. If the string unwinds horizontally from the top of the core with a constant speed of 29.0 cm/s,

(a) What is the speed of the spool's ground-touching point?

(b) What is the angular speed of the spool? (Take the ground point as the instantaneous axis of rotation)

(c) What is the speed of the top point of the spool?

(d) What is the speed of CM of the spool?

(e) What is the speed of the spool?

Explanation / Answer

Part A: It has been given in the question that the spool rolls without slipping. Therefore there is no relative velocity between the spool's found touching point and the ground itself as there is no slipping. Hence the velocity of spool's ground touching point is same as the ground itself. Hence the velocity is zero. Answer= zero Part B: Taking the ground point as the instantaneous axis of rotation let the angular speed of the spool be w. we know that (w)R=v where r is the radial distance and v is velocity It is given that the velocity of the inner core's top point is 29cm/s. And its distance from the ground is (1.5+1)cm = 2.5cm Therefore R= 2.5cm hence using (w)R=V (w)2.5=29 hence w= 11.6 therefore angular velocity= 11.6 radians/sec Answer = 11.6 radians/sec Part C: we know that w=11.6 radians/sec And the distance of the top point of the spool from the ground, X = 2(1.5cm) = 3 cm therefore velocity, v of the top point= (w)X v= (11.6)3 therefore v= 34.8cm/sec Answer= 34.8cm/sec Part D: Center of Mass of the spool is nothing but geometric center of the spool distance of C.M from ground, Y= 1.5cm therefore velocity of C.M is v= (w)Y v= (11.6)1.5= 17.4cm/sec therefore Answer= 17.4cm/sec Part E: When we talk about a solid object's speed we indirectly are commenting on the speed of its center of mass. In other words speed of a ball, bat, rock with indefinite shape means speed of their center of mass. Hence speed of the spool is nothing but speed of its center of mass. Therefore the answer is same as that of Part D Answer+ 17.4cm/sec

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