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A wheel moves in the xy plane in such a way that the location of its center is given by the equations xo = 12t3 and yo = R = 2, where xo and yo are measured in feet and t is measured in seconds. The angular displacement of a radial line measured from a vertical reference line is θ = 8t4, where θ is measured in radians. Determine the velocity of the point P located on the horizontal diameter of the wheel at t = 1.4 s.

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Answer:

the velocity of the point P located on the horizontal diameter of the wheel at t = 1.4 s  is   [tex]P = 104.04 \hat{i} -314.432 \hat{j}[/tex]

Explanation:

The free-body  diagram below shows the interpretation of the question; from the diagram , the wheel that is rolling in a clockwise directio will have two velocities at point P;

  • the peripheral velocity that is directed downward [tex](-V_y)[/tex] along the y-axis
  • the linear velocity [tex](V_x)[/tex] that is directed along the x-axis

Now;

[tex]V_x = \frac{d}{dt}(12t^3+2) = 36 t^2[/tex]

[tex]V_x = 36(1.7)^2\\\\V_x = 104.04\ ft/s[/tex]

Also,

[tex]-V_y = R* \omega[/tex]

where [tex]\omega[/tex](angular velocity) = [tex]\frac{d\theta}{dt} = \frac{d}{dt}(8t^4)[/tex]

[tex]-V_y = 2*32t^3)\\\\\\-V_y = 2*32(1.7^3)\\\\-V_y = 314.432 \ ft/s[/tex]

∴ the velocity of the point P located on the horizontal diameter of the wheel at t = 1.4 s  is   [tex]P = 104.04 \hat{i} -314.432 \hat{j}[/tex]

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