Question

A link \( OA \) of length 200 mm is rotating counterclockwise about \( O \) in the \( x \)-\( y \) plane with a constant angular velocity of 100 rad/s, as shown in the figure. The absolute value of the \( x \)-component of the linear velocity (in m/s) of point \( A \) at the instant shown in the figure is .............. 

Hint:

When calculating the linear velocity of a rotating object, use the formula \( v = r \cdot \omega \), and remember to project the velocity onto the required direction (in this case, the \( x \)-axis).

Answer 1

Correct Answer: 9.8

The linear velocity \( v \) of point \( A \) can be calculated using the formula: \( v = \omega \times r \), where \( \omega = 100 \, \text{rad/s} \) is the angular velocity, and \( r = 200 \, \text{mm} = 0.2 \, \text{m} \) is the distance from the origin \( O \) to point \( A \).

The total linear velocity is given by:
\( v = 100 \times 0.2 = 20 \, \text{m/s} \)

The \( x \)-component of the velocity, \( v_x \), is given by:
\( v_x = v \times \sin(\theta) \), where \( \theta = 30^\circ \).

Substituting the values:
\( v_x = 20 \times \sin(30^\circ) = 20 \times 0.5 = 10 \, \text{m/s} \)

The absolute value of the \( x \)-component of the linear velocity is \( 10 \, \text{m/s} \).

Confirming the provided range of \( 9.8 \leq v_x \leq 9.8 \), we find that the calculated value \( 10 \, \text{m/s} \) is slightly outside the range. The problem may have an incorrect expected range; however, the calculated result is accurate.