Answer to Question #257825 in Physics for THOMAS

Question

A roll of toilet paper is held by the first piece and allowed to unfurl as shown in the diagram to the right. The roll has an outer radius R = 6.0 cm, an inner radius r = 1.8 cm, a mass m = 200 g, and falls a distance s = 3.0 m. Assuming the outer diameter of the roll does not change significantly during the fall, determine…

1. the tension in the sheets
2. the translational acceleration of the roll
3. the angular acceleration of the roll
4. the final translational speed of roll
5. the final angular speed of the roll

Answer 1

The translational and rotational motion give us:

$$ma=mg-T,\\ RT=\frac12m(R^2+r^2)\frac aR,\\\space\\ T=\frac{(R^2+r^2)mg}{3R^2+r^2}=0.691\text{ N}.$$

The linear acceleration is

$$a=g-\frac Tm=6.34\text{ m/s}^2.$$

The angular acceleration is

$$\alpha=\frac aR=106\text{ rad/s}^2.$$

The final linear speed is

$$v=\sqrt{v^2+2ad}=6.17\text{ m/s}.$$

The angular speed:

$$\omega=v/R=103\text{ rad/s}.$$