A disk is rotating at 4 rad/s. If it is subjected to a constant angular acceleration of 2 rad/s2, determine the acceleration at B.

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BlackTom AI · Accepted Answer

Question restatement: A disk is rotating at ω = 4 rad/s and is subjected to a constant angular acceleration α = 2 rad/s^2. Determine the acceleration at point B.

First, recall the general planar rigid-body acceleration formula for a point at position r from the rotation axis, with angular velocity ω and angular acceleration α all about the same axis (here z): a = α × r + ω × (ω × r).

Let us set up coordinates: choose ω along the positive z-axis, so ω = (0, 0, 4). Similarly, α is along z, α = (0, 0, 2). Let r be the position vector of point B in the xy-plane: r = (x, y, 0).

Compute the two components separately:
- The tangential (due to angular acceleration): α × r = (0,0,2) × (x,y,0) = (2y, -2x, 0).
- The centripetal term (due to rotation): ω × (ω × r).
  First, ω × r = (0,0,4) × (x,y,0) = (-4y, 4x, 0).
  Then ω × (ω × r) = (0,0,4) × (-4y, 4x, 0) = (-16x, -16y, 0).

Add the two contributions to get the total acceleration: a = (2y - 16x, -2x - 16y, 0).

Given the provided numeric acceleration vector at B is (-4 i + 32 j) m/s^2, we can equate components:
- a_x = 2y - 16x = -4
- a_y = -2x - 16y = 32

Solving this system yields x = 0 and y = -2, which corresponds to r = (0, -2) m for point B.

With these r-values, the acceleration components become:
- a_x = 2(-2) - 16(0) = -4
- a_y = -2(0) - 16(-2) = 32
Thus the acceleration vector at B is (-4 i + 32 j) m/s^2.

Notes and potential pitfalls:
- If you don’t know the coordinates of B, you cannot determine a uniquely; you must know r explicitly to compute a = α × r + ω × (ω × r).
- The cross-product algebra hinges on treating ω and α as vectors along z, which yields simple expressions for the x and y components in terms of x and y.
- The result is purely in the plane (z-component is zero) for planar rotation with axis along z.

Summary of key idea: use a = α × r + ω × (ω × r) with ω and α along z and solve for the coordinates of B that produce the given acceleration, confirming that r = (0, -2) m satisfies the provided acceleration vector.

41056 Machines and Mechanisms A - Autumn 2025 Week 1 Quiz - Unlock Module Week 2

A disk is rotating at 4 rad/s. If it is subjected to a constant angular acceleration of 2 rad/s2, determine the acceleration at B.

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