A block of mass 12 kg starts from rest and

slides a distance of 8 m down an inclined
plane making an angle of 40◦
with the horizontal. The coefficient of sliding friction between block and plane is 0.4.
The acceleration of gravity is 9.8 m/s^2

What is the net force on the block along the
incline?

What is the speed of the block after sliding
8 m?

What would be its speed if friction were negligible?

To find the net force on the block along the incline, we can first calculate the force due to gravity. The force due to gravity can be found using the formula:

F_gravity = m * g

where:
m = mass of the block = 12 kg
g = acceleration due to gravity = 9.8 m/s^2

Plugging in these values, we get:

F_gravity = 12 kg * 9.8 m/s^2
= 117.6 N

Next, we can calculate the force of friction using the formula:

F_friction = μ * F_normal

where:
μ = coefficient of sliding friction = 0.4
F_normal = perpendicular force to the inclined plane

The normal force can be calculated as:

F_normal = m * g * cosθ

where:
θ = angle of the inclined plane = 40°

Plugging in the values, we get:

F_normal = 12 kg * 9.8 m/s^2 * cos(40°)

Now, we can calculate the force of friction:

F_friction = 0.4 * 12 kg * 9.8 m/s^2 * cos(40°)

The net force along the incline can be calculated using:

F_net = F_gravity - F_friction

Plugging in the values, we get:

F_net = 117.6 N - (0.4 * 12 kg * 9.8 m/s^2 * cos(40°))

To find the speed of the block after sliding 8 m, we can use the kinematic equation:

v² = u² + 2as

where:
v = final velocity
u = initial velocity (which is 0 m/s since the block starts from rest)
a = acceleration (which is the net force divided by the mass)
s = distance traveled (which is 8 m)

The initial velocity u is 0 m/s, so the equation simplifies to:

v² = 2 * (F_net / m) * s

Plugging in the values we calculated earlier, we can find the speed v.

Finally, if friction were negligible, the net force along the incline would only be the force due to gravity. Therefore, the force of friction would be zero, and the net force would be the same as the force due to gravity. Using the formula:

F_net = m * g

we can find the net force when friction is negligible. Using the mass and acceleration due to gravity values, we can calculate the net force. And then, using the same kinematic equation as before, we can find the speed.