Vertical direction of acceleration in a simple pendulum
In a simple pendulum, the bob (mass) is attached to a string or a rod of negligible mass and length. When the bob is displaced from its equilibrium position and released, it oscillates back and forth under the influence of gravity.

1) Vertically upwards acceleration
To determine the value of θ (angle with vertical) for which the acceleration is directed vertically upwards, we need to consider the forces acting on the bob. The two main forces are the tension in the string and the weight of the bob.

When the bob is at its maximum displacement (angular amplitude of 90 degrees), the tension in the string is zero. At this point, the entire weight of the bob acts vertically downwards.

As the bob starts to move back towards the equilibrium position, the tension in the string increases. At some angle θ less than 90 degrees, the tension becomes equal to the weight of the bob. At this point, the net force acting on the bob is zero, resulting in zero acceleration in the vertical direction.

Therefore, for an angular amplitude of 90 degrees, the acceleration is directed vertically upwards when θ is less than 90 degrees. This means that the bob must be on its way back towards the equilibrium position but has not reached it yet.

2) Horizontally directed acceleration
In a simple pendulum, the acceleration is always directed towards the equilibrium position and is perpendicular to the velocity of the bob. Therefore, the acceleration is never directed horizontally.

3) Vertically downwards acceleration
When the bob is at its maximum displacement (angular amplitude of 90 degrees), the acceleration is directed vertically downwards. This is because the tension in the string is zero and the weight of the bob acts vertically downwards.

As the bob moves towards the equilibrium position, the tension in the string increases and the net force acting on the bob becomes non-zero. The acceleration is always directed towards the equilibrium position, which is vertically downwards.

Therefore, for an angular amplitude of 90 degrees, the acceleration is directed vertically downwards when θ is greater than 90 degrees. This means that the bob has already passed the equilibrium position and is moving away from it.

In summary, the acceleration in a simple pendulum is directed vertically upwards when θ is less than 90 degrees, vertically downwards when θ is greater than 90 degrees, and never horizontally.