a series of small packages are being moving by athin conveyor belt that passes over a 300mmradiusidler pulley.the belt starts from rest at time t=0 and its speed increses at a constant rate of 150mm/s*2.knowing that the coffient of static friction between the packages and the belt is 0.75.determine the time at which the first package slips.

To determine the time at which the first package slips, we need to analyze the forces acting on the package.

Let's consider the package's motion in two stages:
1. The package remains stationary on the conveyor belt until the maximum static friction is reached.
2. The package starts to slip and slide on the conveyor belt.

Stage 1: The package remains stationary
In this stage, the static friction between the package and the belt prevents the package from slipping. The maximum static friction force (f_static_max) is given by the equation:
f_static_max = coefficient_of_static_friction * normal_force

The normal force is the force acting perpendicularly to the contact surface, which is equal to the weight of the package:
normal_force = package_mass * gravitational_acceleration

Stage 2: The package slips
Once the maximum static friction force is reached, the package starts to slip. At this point, the friction force changes from static friction to kinetic friction. The kinetic friction is given by the equation:
f_kinetic = coefficient_of_kinetic_friction * normal_force

Now, let's calculate the time at which the first package slips:

1. Calculate the static friction force when the package is stationary:
f_static_max = 0.75 * (package_mass * gravitational_acceleration)

2. Calculate the acceleration of the package:
acceleration = (150 mm/s^2)

3. Calculate the maximum speed the belt can reach before the package starts to slip:
max_speed = (acceleration * time_at_slipping)

4. At the time the package starts to slip, the maximum speed of the belt is equal to the speed of the belt when the package slips:
max_speed = (acceleration * time_at_slipping)

5. Calculate the distance traveled by the belt before the package starts to slip:
distance = 2 * pi * (300 mm)

6. Substitute the distance and maximum speed into the equation above to solve for time_at_slipping:

(acceleration * time_at_slipping) = (2 * pi * 300 mm)

Rearrange the equation to solve for time_at_slipping:

time_at_slipping = (2 * pi * 300 mm) / acceleration

Calculate the value using the given acceleration.

So, based on the given information, you can determine the time at which the first package slips by substituting the values into the equation provided.