To solve the problem of finding the velocity and acceleration of the center of gravity (G) of the disc with a semi-circular hole, we need to apply some principles of rotational motion and kinematics. The key here is to relate the angular velocity and angular acceleration of the disc to the linear velocity and acceleration of its center of gravity.

Understanding the Relationship Between Angular and Linear Motion

When an object rolls without slipping, there is a direct relationship between its angular motion and its linear motion. For a disc, the following equations hold:

• Linear Velocity (v): This is related to the angular velocity (ω) by the equation: v = r * ω, where r is the radius of the disc.
• Linear Acceleration (a): This is related to the angular acceleration (α) by the equation: a = r * α.

Given Data

From the problem, we have:

• Angular velocity, ω = 5 rad/s
• Angular acceleration, α = 3 rad/s²

Calculating the Radius of the Disc

To proceed, we need to determine the radius of the disc. Since the problem does not specify the radius, we can denote it as r. The equations will remain general until we substitute a specific value for r.

Finding the Linear Velocity of G

Using the relationship for linear velocity, we can express the velocity of the center of gravity (G) as follows:

Substituting the given angular velocity:

Finding the Linear Acceleration of G

Next, we calculate the linear acceleration of G using the angular acceleration:

Substituting the given angular acceleration:

Summary of Results

At this point, we have derived the expressions for the linear velocity and acceleration of the center of gravity (G) of the disc:

• Linear Velocity: v = 5r (in m/s)
• Linear Acceleration: a = 3r (in m/s²)

To find specific numerical values for velocity and acceleration, you would need to know the radius of the disc. However, the relationships we derived hold true for any radius. This approach illustrates how angular motion translates into linear motion for rolling objects.