Question

A sledgehammer is being used to break up concrete chunks. To use the sledgehammer, the construction worker is lifting the sledge over their head and swinging it downward to strike. At one particular time during the swing, the sledgehammer is poised as shown below:

Assume that the sledgehammer is pivoting around the end of the handle (assumed to be the origin with a position of (0, 0, 0) m. One point on the sledgehammer is measured to find its velocity. Point A is located at a distance vector R1 = (0.62, 0.07) m. Point A has a velocity of (2.6, -18.8) m/s. The measured rotation vector of the sledgehammer at that moment is (0, 0, 4.0) rad/s, find the net magnitude of the velocity of the center of rotation of the sledgehammer. (Note: Answer in m/s)

Answer 1

The net magnitude of the velocity of the center of rotation of the sledgehammer is 0 m/s because it is the pivot point and thus stationary relative to the hammer.

Step-by-step explanation:

To find the net magnitude of the velocity of the center of rotation of the sledgehammer, we use the relationship between linear velocity (V), angular velocity (ω), and the radius (r) in rotational motion. Since point A located at distance vector R1 with coordinates (0.62, 0.07)m has a velocity of (2.6, -18.8)m/s, we know the following:

V = ω × r

The angular velocity vector given is (0, 0, 4.0)rad/s, indicating the rotation is around the z-axis. To find the velocity at the center of rotation, we can set up the equation using the known linear velocity of point A:

¶ VA = ω × r1

We can cross-multiply the rotation vector with the position vector R1 to find the velocity vector of point A. The result should match the measured velocity vector (2.6, -18.8)m/s.

Now, in order to find the velocity of the center of rotation, we note that the center of rotation is stationary relative to the sledgehammer itself, which is moving with angular velocity ω. Thus, we could use the magnitude of velocity VA and the length of the radius r1 to calculate the angular velocity. However, we already have the angular velocity, and we can see that any point on the axis of rotation will have a linear velocity of zero. Therefore, the net magnitude of the velocity of the center of rotation is 0 m/s, as it is the pivot point.