Problem 5: Toilet Paper Problem ow long does it take a toilet paper roll to hit

Question

Problem 5: Toilet Paper Problem ow long does it take a toilet paper roll to hit the fHoor, if it was released from a height of 0.73 m, while you are holding onto the first sheet (as shown in the figure)? The roll has an inner radius of 2.7 cm an outer radius of 6.1 cm, and a mass of 274 g. Assume that its mass and outer radius dont change significantly over the course of the fall. a) Find an expression for the acceleration of the roll depending only on the two given radi and g, through a combination of linear and rotational dynamics, then use kinematics to calculate the ti b) Find the final speed of the roll through energy conservation Note: the moment of inertia of a hollow cylinder is M(R1 + R), where R1 is the internal radius, me. and R2 the external one.

Explanation / Answer

Prob 5:

force equation is given as

mg - T = ma

T = mg - ma eq-1

Torque equation is given as

T R2 = Ia/R2

T = Ia/R22

using eq-1

mg - ma =  Ia/R22

mg - ma =  (0.5) m (R12 + R22)a/R22

2gR22 - 2aR22 = (R12 + R22)a

a = 2gR22 /(R12 + 3R22)

Vo = initial velocity = 0 m/s

Vf = final velocitty

t = time taken

using the equation

Vf = Vo + at

Vf = 0 + 2gR22 t/(R12 + 3R22)

Vf = 2gR22 t/(R12 + 3R22)

b)

using conservation of energy

kinetic energy gained + rotational KE = Potential energy lost

(0.5) m v2 + (0.5) I w2 = mgh

(0.5) m v2 + (0.25) m (R12 + R22) (v/R2)2 = mgh

(0.5) v2 + (0.25) ((0.027)2 + (0.061)2) (v/0.061)2 = 9.8 x 0.73

v = 3 m/s

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