A 1750 kg van is travelling down a 10 degrees inclined road

at a speed of 6 m/s. If the driver brakes suddenly causing the
wheels to lock, determine the distance L his tyres will skid on
the road. The coefficient of kinetic friction between the
wheels and the road is k=0.5.
(a) Draw the free body diagram of the forces. [4 marks]
(b) Use the equation of equilibrium to calculate the
Friction Force Fk.
[6 marks]
(b) Apply the principle of Work and Energy to find L?
[10 marks]

didn't I just respond to this?

(a) To draw the free body diagram of the forces acting on the van on the inclined road, we consider the forces acting in the vertical and horizontal directions.

Vertical forces:
- Weight (mg): This is the force acting vertically downwards and can be calculated by multiplying the mass of the van (m = 1750 kg) by the acceleration due to gravity (g = 9.8 m/s^2).

Horizontal forces:
- Normal force (N): This is the force acting perpendicular to the inclined road. It can be calculated by taking the component of the weight in the perpendicular direction, which is given by N = mg * cos(10°).
- Friction force (Fk): This is the force acting parallel to the inclined road due to the friction between the wheels and the road. We need to calculate this force using the equation of equilibrium.

(b) Using the equation of equilibrium, the sum of the forces in the horizontal direction is zero because the van is not accelerating horizontally. Therefore, the equation becomes:

Sum of forces in the horizontal direction = 0
Fk = N * μk

Substituting the given values:
μk = 0.5 (coefficient of kinetic friction)
N = mg * cos(10°) (normal force)

We can solve for sing the values of m, g, and the angle of inclination.

(c) To find the distance L, we can use the principle of Work and Energy. The work done by the friction force in stopping the van is equal to the change in kinetic energy of the van.

The work done by the friction force is given by:
Work = Fk * L

The change in kinetic energy is given by:
ΔKE = 0.5 * m * v^2

Since the van is brought to rest, the final kinetic energy is zero:
ΔKE = 0 - 0.5 * m * v^2

Equating the work done by the friction force to the change in kinetic energy, we can solve for L:

Fk * L = -0.5 * m * v^2

Substituting the known values of Fk, m, and v, we can solve for L.

(a) To draw the free body diagram of the forces acting on the van, we will consider the forces acting on the van on the inclined road.

1. Weight (W): It acts vertically downwards and can be calculated using the formula W = mg, where m is the mass of the van (1750 kg) and g is the acceleration due to gravity (9.8 m/s^2).
2. Normal force (N): It acts perpendicular to the surface of the road and counterbalances the weight of the van. It can be calculated using the formula N = mg*cos(theta), where theta is the angle of the inclined road (10 degrees).
3. Friction force (Fk): It acts parallel to the surface of the road and opposes the motion of the van. The formula to calculate the friction force is Fk = N * uk, where uk is the coefficient of kinetic friction (0.5).
4. Tension force (T): It acts in the opposite direction of the movement of the van and is indirectly applied by the brakes. However, since the wheels lock and no longer rotate, the tension force can be ignored for this case.

So, the free body diagram will have a downward arrow representing the weight (W), an upward arrow representing the normal force (N), and a backward arrow representing the friction force (Fk).

(b) Using the equation of equilibrium, we can calculate the friction force (Fk) acting on the van.

Since the van is in equilibrium in the horizontal direction (no acceleration), the sum of all horizontal forces must be zero.

In this case, the only horizontal force is the friction force (Fk). So, we can write the equation as:

Fk = 0

Since the wheels lock and the van is not accelerating horizontally, the friction force balances the horizontal component of the weight of the van.

So, the friction force (Fk) can be calculated as:

Fk = W*sin(theta)

Substituting the values:

W = mg = (1750 kg) * (9.8 m/s^2)
theta = 10 degrees

Fk = (1750 kg) * (9.8 m/s^2) * sin(10 degrees)

Solving this will give the value of the friction force (Fk).

(c) To find L (the distance the tires will skid on the road), we will apply the principle of Work and Energy.

The work done on an object is given by the formula:

Work = Force * Distance * cos(angle)

In this case, the work done against the friction force (Fk) will be equal to the change in kinetic energy of the van.

The work done against friction force can be calculated as:

Work = Fk * L * cos(180 degrees) (since the force and displacement are in opposite directions)

The change in kinetic energy can be calculated using the formula:

Change in Kinetic Energy = 0.5 * m * (final velocity^2 - initial velocity^2)

Since the van comes to rest, the final velocity is 0. Therefore, we can write:

0.5 * m * (0 - (6 m/s)^2) = Fk * L * cos(180 degrees)

Now, substitute the value of Fk calculated from part (b) and solve for L to find the distance the tires will skid on the road.