A rat with mass 0.5 kg is sliding down an inclined plane. The plane is at an angle of 34⁰ with respect to the ground, and the coefficient of friction between the rat and the plane is 0.2. Simultaneously, a cat is giving the rat a push UP the hill, with a force equal to 0.4 N. Is the rat accelerating down the hill? Show why or why not, using Newton’s second law. If the rat is accelerating, calculate his acceleration.

Wr = m*g = 0.5kg * 9.8N/kg = 4.9 N = Wt.

of the rat.

Fp = 4.9*sin34 = 2.740 N. = Force to
the plane.
Fn = 4.9*cos34 = 4.062 N. Normal = Force perpendicular to the plane.

Fk = u*Fn = 0.2 * 4.062 = 0.8125 N.

Fc-Fp-Fk = m*a
0.4-2.74-0.8125 = m*a
m*a = -3.15
a = -3.15/m = -3.15/0.5= -7.30 m/s^2.

The negative acceleration means that the rat is accelerating DOWN the plane
in spite of the upward force exerted by the cat. The opposing forces are greater than the force of the cat.

To determine if the rat is accelerating down the hill, we need to consider the forces acting on the rat and apply Newton's second law. Newton's second law states that the net force acting on an object is equal to the product of its mass and acceleration.

First, let's identify the forces acting on the rat:
1. The force of gravity (weight) pulling the rat downwards.
2. The normal force exerted by the inclined plane perpendicular to it.
3. The force of friction opposing the motion.
4. The force exerted by the cat pushing the rat up the hill.

Now, let's calculate the gravitational force pulling the rat downwards:
Weight = mass * acceleration due to gravity
Weight = 0.5 kg * 9.8 m/s^2
Weight = 4.9 N

Next, we need to calculate the normal force exerted by the inclined plane. Since the plane is inclined, the normal force can be determined by decomposing the weight vector along the inclined plane:
Normal force = weight * cos(angle of inclination)
Normal force = 4.9 N * cos(34°)
Normal force ≈ 4.07 N

The force of friction can be calculated using the equation:
Force of friction = coefficient of friction * normal force
Force of friction = 0.2 * 4.07 N
Force of friction ≈ 0.814 N

Now, let's determine if the rat is accelerating down the hill by analyzing the net force acting on it:
Net force = force pushing up - force of friction - force due to gravity
Net force = 0.4 N - 0.814 N - 4.9 N
Net force ≈ -5.314 N

Since the net force has a negative value of -5.314 N, it means that there is a net force acting in the downward direction, and the rat is indeed accelerating down the hill.

Finally, to calculate the rat's acceleration, we can use Newton's second law by rearranging the formula:
Net force = mass * acceleration
-5.314 N = 0.5 kg * acceleration
Acceleration ≈ -10.63 m/s^2

The rat's acceleration is approximately -10.63 m/s^2, indicating that it is accelerating down the hill. The negative sign indicates that the acceleration is in the opposite direction of the positive direction chosen (downwards in this case).