The triangular plate rotates about a fixed axis through point O. The angular velocity is ω =9 rad/s anti-clockwise and the angular acceleration is α = 2 rad/s2 clockwise. The lengths ofthe vertical side is h = 30 cm and the length of the horizontal side is b = 45 cm.What are the magnitudes of velocity and acceleration of point A at the instant shown
Question
The triangular plate rotates about a fixed axis through point O. The angular velocity is ω =9 rad/s anti-clockwise and the angular acceleration is α = 2 rad/s2 clockwise. The lengths ofthe vertical side is h = 30 cm and the length of the horizontal side is b = 45 cm.What are the magnitudes of velocity and acceleration of point A at the instant shown
Solution
To find the velocity and acceleration of point A, we need to use the formulas for rotational motion.
- Velocity (v) of a point in a rotating body is given by the formula v = rω, where r is the distance of the point from the axis of rotation and ω is the angular velocity.
In this case, the distance of point A from the axis of rotation (O) is the hypotenuse of the triangle formed by the sides h and b. Using the Pythagorean theorem, we find r = sqrt(h^2 + b^2) = sqrt((30 cm)^2 + (45 cm)^2) = 54 cm = 0.54 m.
So, the velocity of point A is v = rω = 0.54 m * 9 rad/s = 4.86 m/s.
- The tangential acceleration (a_t) of a point in a rotating body is given by the formula a_t = rα, where α is the angular acceleration.
The angular acceleration is given as 2 rad/s^2, but it's in the opposite direction to the angular velocity. In terms of magnitude, we ignore the direction, so we can use the same formula.
So, the tangential acceleration of point A is a_t = rα = 0.54 m * 2 rad/s^2 = 1.08 m/s^2.
- The radial (or centripetal) acceleration (a_r) of a point in a rotating body is given by the formula a_r = rω^2.
So, the radial acceleration of point A is a_r = rω^2 = 0.54 m * (9 rad/s)^2 = 43.74 m/s^2.
- The total acceleration (a) of a point in a rotating body is given by the formula a = sqrt(a_t^2 + a_r^2).
So, the total acceleration of point A is a = sqrt((1.08 m/s^2)^2 + (43.74 m/s^2)^2) = 43.77 m/s^2.
So, the magnitude of the velocity of point A is 4.86 m/s and the magnitude of the acceleration is 43.77 m/s^2.
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