The magnitude of the tension force that the rod exerts on the paper roll as it u | TestHub

Passage / Comprehension

A large $16.0$ kg roll of paper with radius $R = 18.0$ cm rests against the wall and is held in place by a smooth axis attached to a rod through the centre of the roll as shown in the figure. The moment of inertia of the paper roll about the axis is $0.360$ kg m $^{2}$ . The other end of the axis is attached by a frictionless hinge to the wall such that the axis makes an angle of $3 7^{\circ}$ with the wall. The weight of the axis is negligible. The coefficient of kinetic friction between the paper and the wall is $μ_{k} = 1/3$ . A constant vertical force $F = 400$ N is applied to the paper, and the paper unrolls. ( $g = 10$ m/s $^{2}$ )

The magnitude of the tension force that the rod exerts on the paper roll as it unrolls is :

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Answer:

Solution:

From the force balance on the roll (Q.11): $N = T sin θ$ , $f_{k} = μ_{k} N$ , and $F + m g + f_{k} = T cos θ$ .

$T = \frac{F + m g}{c o s θ - μ _{k} s i n θ} = \frac{400 + 160}{\frac{4}{5} - \frac{1}{3} \cdot \frac{3}{5}} = \frac{560}{\frac{3}{5}} = \frac{2800}{3}$ N - (D)

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