Why Does the Ball Lag Behind the Falling Board at Angles Less Than 35.3 Degrees?

[es19]

problem: a common demonstration, consists of a ball resting at one end of a uniform board of length =L, hinged at the other end, and elevated at an angle theta. a light cup is attached to the board at a distance d from the hinge so that it will catch the ball when the support stick is suddenly removed, which means that d=Lcos(theta) show that the ball will lag behind the falling board when theta is less than 35.3 degrees.

ive found the free acceleration of the end of the beam and the center of mass of the beam which are 3/2g and 3/4g respectively but am not sure where to go from there??

 

[Tide]

Shouldn't the acceleration at the end of the board depend on the angle of elevation?

 

[quicknote]

First of all, great job on finding the free acceleration of the end of the beam and the center of mass! To show that the ball will lag behind the falling board when theta is less than 35.3 degrees, we can use the equation for the angular acceleration of the board, which is given by alpha = (2/3)(g/L)sin(theta). This equation tells us that the angular acceleration of the board is directly proportional to the sine of theta, meaning that as theta decreases, the angular acceleration also decreases.

Now, let's consider the motion of the ball. The ball will experience two forces: the gravitational force pulling it down and the normal force from the cup pushing it up. When the support stick is removed, the ball will initially stay in place due to inertia, but as the board starts to fall, the normal force from the cup will also decrease as the distance between the ball and the cup increases. This means that the net force on the ball will be in the downward direction, causing it to lag behind the falling board.

To determine the angle at which the ball will start to lag behind, we can set the net force on the ball equal to zero. This occurs when the normal force from the cup is equal to the gravitational force, which can be expressed as:

mg = N = m(3/4)gcos(theta)

Solving for theta, we get:

theta = arccos(4/3) ≈ 35.3 degrees

Therefore, when theta is less than 35.3 degrees, the normal force from the cup will not be enough to counteract the gravitational force, causing the ball to lag behind the falling board. This demonstrates the importance of understanding the forces acting on an object in order to predict its motion accurately. Keep up the good work!