A uniform 2.00-m-long stick is raised at an angle of 30° to the horizontal above a sheet of ice. The bottom end of the stick rests on the ice. The stick is released from rest. The bottom end of the stick remains in contact with the ice at all times. How far will the bottom end of the stick have traveled during the time the rest of the stick is falling to the ice? Assume that the ice is frictionless.

Question

A uniform 2.00-m-long stick is raised at an angle of 30° to the horizontal above a sheet of ice. The bottom end of the stick rests on the ice. The stick is released from rest. The bottom end of the stick remains in contact with the ice at all times. How far will the bottom end of the stick have traveled during the time the rest of the stick is falling to the ice? Assume that the ice is frictionless.

Answer 1

There are no horizontal forces on the stick so the center of mass of the stick will not move sideways.

Therefore the middle of the stick will end up exactly below where it started.
The end of the stick will move a little because now the stick is flat on the ice.
It will be one meter from the center instead of 1 cos 30 horizontally.

Answer 2

To determine the distance traveled by the bottom end of the stick while the rest of the stick falls, we need to consider the motion of the stick.

First, we can find the time it takes for the top end of the stick to reach the ice. We know that the acceleration due to gravity is 9.8 m/s², and the initial velocity is 0 m/s since the stick is released from rest. The formula for the time it takes to fall can be derived from the kinematic equation:

d = (1/2) * a * t²

where d is the vertical distance traveled, a is the acceleration due to gravity, and t is the time taken. Rearranging the equation, we get:

t = sqrt(2 * d / a)

Since the stick has a length of 2.00 m, the top end will fall a distance of 2.00 m. Plugging in the values, we find:

t = sqrt(2 * 2.00 m / 9.8 m/s²)
t ≈ 0.64 s

Therefore, it takes approximately 0.64 seconds for the top end of the stick to reach the ice.

Now, we can find the horizontal distance traveled by the bottom end of the stick during this time. Since the stick forms a right triangle with the horizontal surface, the horizontal distance is given by:

d_horizontal = d_vertical * tan(angle)

where d_horizontal is the horizontal distance traveled, d_vertical is the vertical distance traveled (which is 2.00 m), and angle is the angle formed between the stick and the horizontal surface (which is 30°).

Plugging in the values, we get:

d_horizontal = 2.00 m * tan(30°)
d_horizontal ≈ 1.16 m

Therefore, the bottom end of the stick will travel approximately 1.16 meters horizontally while the rest of the stick falls to the ice.