Question

The disk rotates about a fixed axis through point O with a clockwise angular velocity ω0 = 16 rad/s and a counterclockwise angular acceleration α0 = 5.6 rad/s2 at the instant under consideration. The value of r is 280 mm. Pin A is fixed to the disk but slides freely within the slotted member BC. Determine the velocity and acceleration of A relative to slotted member BC and the angular velocity and angular acceleration of BC. The relative velocity and acceleration are positive if they point from B to C and negative if they point from C to B. The angluar velocity and angular accelerations are positive if counterclockwise, negative if clockwise.

Answer 1

Velocity of A relative to BC: v_A_relative = 4.48 m/s from C to B
Acceleration of A relative to BC: a_relative = 1.568 m/s² from B to C
Angular velocity of BC: ω_BC = 16 rad/s (counterclockwise, therefore positive)
Angular acceleration of BC: α_BC = 0 rad/s² (no information suggests it's changing at the instant)

Step-by-step explanation:

To analyze the motion of pin A and the slotted member BC, we must recognize that there are two components to both the velocity and acceleration of pin A: the component due to the disk's rotation and the component due to the motion relative to the slotted member BC. Angle measurements are presumed to be in radians, time in seconds, distances in meters, velocities in meters per second (m/s), and accelerations in meters per second squared (m/s²).

Given:
ω0 (angular velocity of the disk) = 16 rad/s (clockwise, therefore negative)
α0 (angular acceleration of the disk) = 5.6 rad/s² (counterclockwise, therefore positive)
r (radius) = 280 mm = 280 x 10^-3 m = 0.280 m

1. Determine the velocity of pin A relative to the disk.

Since we are looking at the instant where the disk's rotational velocity and acceleration are given, we can use the tangential velocity formula: v_t = r * ω

v_t = r * ω0
v_t = 0.280 m * 16 rad/s (since the disk rotates clockwise, the direction of v_t is towards B)
v_t = 4.48 m/s

So, the tangential velocity of A relative to O is 4.48 m/s directed from C towards B (since that's the direction of rotation).

2. Determine the tangential acceleration of pin A.

For the tangential acceleration, we use the formula: a_t = r * α

a_t = 0.280 m * 5.6 rad/s²
a_t = 1.568 m/s²

Since α0 is positive (counterclockwise), and the acceleration is tangential to the path of pin A, it will be directed from B towards C.

3. Determine the centripetal (radial) acceleration of pin A.

Centripetal acceleration is given by: a_c = r * ω^2

a_c = 0.280 m * (16 rad/s)²
a_c = 0.280 m * 256 rad²/s²
a_c = 71.68 m/s²

The centripetal acceleration always points towards the center of rotation, which, in this case, is point O of the disk. Therefore, it will point from A towards O perpendicularly to the tangential acceleration.

4. Determine the total relative acceleration of pin A with respect to BC.

Since pin A is in rotational motion, it does not have relative acceleration with respect to the slot except for the tangential acceleration that it has because of the disk's angular acceleration. Therefore, the total relative acceleration of A with respect to BC is the tangential acceleration:

a_relative = a_t = 1.568 m/s² from B to C.

There is no additional acceleration component to consider in the direction of BC.

5. Determine the angular velocity and acceleration of BC.

From the perspective of the slotted member BC, point A appears to move in a straight line. Since all points on the rod would have the same angular velocity, ω_BC can be calculated using the velocity v_t of point A and the distance r from point O.

ω_BC = v_A_relative / r
ω_BC = 4.48 m/s / 0.280 m
ω_BC = 16 rad/s (same as the disk's angular velocity ω0)

The angular velocity of BC is 16 rad/s relative to point O. Since the system is viewed at an instant and there is no additional information about the change in ω_BC, we assume that the angular acceleration of BC α_BC is zero.

Finally, we can summarize:
Velocity of A relative to BC: v_A_relative = 4.48 m/s from C to B
Acceleration of A relative to BC: a_relative = 1.568 m/s² from B to C
Angular velocity of BC: ω_BC = 16 rad/s (counterclockwise, therefore positive)
Angular acceleration of BC: α_BC = 0 rad/s² (no information suggests it's changing at the instant)

Answer 2

V = 4.48m/s a = 1.57m/s²

Step-by-step explanation:

ω₀ = 16rad/s

α₀ = 5.6rad/s²

r = 280mm = 0.28m

a = ?

v = ?

Angular velocity (ω₀) = velocity of acceleration / length of path

ω₀ = v / r

V = ω₀ * r

V = 16 * 0.28

V = 4.48m/s

Acceleration = ?

Angular acceleration α₀ = angular velocity (ω) / time take (t)

α₀ = ω / t .... equation i

But acceleration (a) = velocity (v) / time (t)

a = v / t

t = v / a

Put t = v / a into equation i

α₀ = ω / (v / a)

α₀ = ω * a / v

α * v = ω * a

a = (α * v) / ω

a = (5.6 * 4.48) / 16

a = 1.568m/s²

a = 1.57m/s²