Two packages at UPS start sliding down the 20° ramp shown in Figure P8.25. Package A has a mass of 3.5 kg and a ­coefficient of friction of 0.20. Package B has a mass of 10 kg and a coefficient of friction of 0.15. How long does it take package A to reach the bottom?

To find the time it takes for package A to reach the bottom of the ramp, we need to use Newton's second law of motion and the equations of motion.

First, let's find the net force acting on package A:

The force of gravity acting on package A can be calculated using the mass and acceleration due to gravity. The formula for calculating the force of gravity is:

Force of gravity = mass * acceleration due to gravity

Given that the mass of package A is 3.5 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the force of gravity on package A:

Force of gravity = 3.5 kg * 9.8 m/s^2 = 34.3 N

Next, we need to calculate the frictional force acting on package A. The formula for calculating the frictional force is:

Frictional force = coefficient of friction * normal force

The normal force is the component of the force of gravity perpendicular to the ramp. It can be calculated using trigonometry:

Normal force = Force of gravity * cos(angle of the ramp)

Given that the angle of the ramp is 20°, we can calculate the normal force:

Normal force = 34.3 N * cos(20°) ≈ 31.4 N

Now we can calculate the frictional force:

Frictional force = 0.20 * 31.4 N ≈ 6.3 N

The net force acting on package A is the difference between the force of gravity and the frictional force:

Net force = Force of gravity - Frictional force = 34.3 N - 6.3 N = 28 N

Now we can use Newton's second law of motion to find the acceleration of package A:

Net force = mass * acceleration

28 N = 3.5 kg * acceleration

Solving for acceleration, we get:

acceleration = 28 N / 3.5 kg ≈ 8 m/s^2

Finally, we can use the equations of motion to find the time it takes for package A to reach the bottom of the ramp. The equation for calculating the time is:

Time = Square root of (2 * distance / acceleration)

We need to know the distance traveled by package A to use this equation. If the length of the ramp is given, we can use trigonometry to find the distance:

Distance = length of the ramp * sin(angle of the ramp)

Without knowing the length of the ramp, we cannot calculate the exact time it takes for package A to reach the bottom.