Question

calculate the linear acceleration
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Answer 1

Answer on Question#52853 - Physics - Mechanics - Kinematics - Dynamics

5/ (20 pts) A yo-yo of mass M, rotational inertia I, and inner and outer radii r and R, is gently pulled by a string with tension T on a rough floor as shown in Figure 1. Calculate the linear acceleration of the yo-yo.

Figure 1

Solution:

Equation of motion, written about the point of contact with the surface, is given by

$$I _ {c} \varepsilon = T \cdot (R - r),$$

where $I_{c}$ – is the moment of inertia about the point of contact, $\varepsilon$ – is the angular acceleration, $T \cdot (R - r)$ – is the torque of the force $T$ about point of the contact. According to the parallel axis theorem the moment of inertia about the point of contact is given by

$$I _ {c} = I + M \cdot R ^ {2},$$

Therefore, the angular acceleration is given by

$$\varepsilon = \frac {T \cdot (R - r)}{I _ {c}} = \frac {T \cdot (R - r)}{I + M \cdot R ^ {2}}$$

The linear acceleration $a$ and the angular acceleration are connected by the following relation

$$a = \varepsilon \cdot R$$

Therefore,

$$a = \frac {T \cdot (R - r)}{I + M \cdot R ^ {2}} \cdot R = \frac {T \cdot R \cdot (R - r)}{I + M \cdot R ^ {2}}$$

Answer: $\frac{T \cdot R \cdot (R - r)}{I + M \cdot R^2}$ .

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