A 160mm diameter pipe of mass 6 kg rests on a 1.5 kg plate. The pipe and plate are initially at rest when a force P of magnitude 25 N is applied for 0.75 s. Knowing that coefficeint of static and dynamic friction between the plate and both the pipe and the floor is 0.25, determine (a) whether the pipe slides with respect to the plate, (b) resulting velocity of the pipe and of the plate.

Let's solve the given problem step by step.

Given Data:

Pipe diameter = 160 mm = 0.16 m
Mass of pipe = 6 kg
Mass of plate = 1.5 kg
Force applied (P) = 25 N
Time duration of force = 0.75 s
Coefficient of static and dynamic friction = 0.25
Step 1: Calculate the total system mass
The total mass of the system is the sum of the mass of the pipe and the plate.

Total mass = 6 kg + 1.5 kg = 7.5 kg

Step 2: Calculate the acceleration of the system if both objects move together
Using Newton's second law:

F_net = (Total mass) × (Acceleration)
25 N = 7.5 kg × a
a = 25 / 7.5
a = 3.33 m/s²

Step 3: Determine the friction force between the pipe and the plate
The normal force exerted by the pipe on the plate is equal to its weight:

Normal force = Mass of pipe × g
= 6 kg × 9.81 m/s²
= 58.86 N

Maximum static friction force (f_s) = μ_s × Normal force
= 0.25 × 58.86
= 14.715 N

If the force needed to accelerate the pipe relative to the plate exceeds this friction value, the pipe will slide.

Step 4: Force needed to accelerate the pipe alone
If both objects are moving together with acceleration
𝑎
=
3.33
a=3.33 m/s², the force required to move the pipe alone at that acceleration is:

Force on pipe = Mass of pipe × a
= 6 × 3.33
= 19.98 N

Since 19.98 N > 14.715 N, the pipe will slide with respect to the plate.

Step 5: Calculate the acceleration of the plate
The friction force between the pipe and the plate will now be the dynamic friction force:

Dynamic friction force = μ_d × Normal force
= 0.25 × 58.86
= 14.715 N

The net force on the plate will be:

Net force on plate = Applied force - Friction force
= 25 N - 14.715 N
= 10.285 N

Now calculate the acceleration of the plate:

Acceleration of plate = Net force on plate / Mass of plate
= 10.285 N / 1.5 kg
= 6.857 m/s²

Step 6: Calculate the acceleration of the pipe
The pipe will experience friction in the opposite direction of the applied force:

Acceleration of pipe = Friction force / Mass of pipe
= 14.715 N / 6 kg
= 2.452 m/s²

Step 7: Calculate final velocities
From the kinematic equation:
v = u + at

For the plate:
Initial velocity = 0
v_plate = 0 + (6.857 m/s²) × 0.75 s
v_plate = 5.14 m/s

For the pipe:
Initial velocity = 0
v_pipe = 0 + (2.452 m/s²) × 0.75 s
v_pipe = 1.84 m/s

Step 8: Final Answer
(a) The pipe slides with respect to the plate.
(b) The resulting velocity of the pipe is 1.84 m/s, and the resulting velocity of the plate is 5.14 m/s.