The circular disk rotates about its center O For the instant

The circular disk rotates about its center O. For the instant represented, the velocity of A is v_a= 7.7j in./sec and the tangential acceleration of B is (a_b)_t = -7.7i in./sec^2. Write the vector expressions for the angular velocity w and angular acceleration a of the disk. Use these results to write the vector expression for the acceleration a_c of point C.

Solution

Sol:

Unit vectors i, j, k are attached to the xyz axes.

The velocity of A can be expressed as vA = × r = k × ri = rj where is the angular velocity of the disk about the z axis and r is the radius of the disk.

When V_A = 7.7j and r = 6.7in,

= V_A / r rad/s.

= 7.7/6.7 = 1.1492 rad/s = 1.15 k rad/s

= 0 i +0 j + 1.15 k

The tangential acceleration of B can be described as (aB)t = ×r = k×rj = ri where is the angular acceleration of the disk about the z axis.

When (aB)t =×r

= -7.7 j / 6.7

= 1.5 k rad/s²

= 0 i + 0 j + 1.5 k

The acceleration of C is a_C = ( × rC) + × ( × rC) = (1.15 k * 6.7 ) + 1.15 *(1.15 k *6.7)

aC = 7.705 i + 8.86 j