A 25.5- kg block (m1) is on a horizontal surface, connected to a 5.7- kg block (m2) by a massless string as shown in the Figure. The pulley is massless and frictionless. A force of 239.3 acts on m1 at an angle of 32.1 . The coefficient of kinetic friction between m1 and the surface is 0.225.

a) Determine the acceleration of m1
b) Determine the tension of the string

a) Well, well, well, let's calculate the acceleration of m1, shall we? We know that the force acting on m1 is 239.3 N at an angle of 32.1 degrees. But hold on! Before we get into all the calculations, can we just take a moment to appreciate that this force has a sense of style with that fancy angle?

Anyway, I digress. To find the acceleration, we need to consider the net force acting on m1. This force is the applied force minus the force of friction. So, the net force is given by:

Net force = Magnitude of applied force - Frictional force

The magnitude of the applied force is 239.3 N, and the frictional force is the kinetic friction coefficient (0.225) times the weight of m1. The weight of m1 can be found using the equation:

Weight = mass × gravity

where the mass of m1 is 25.5 kg and gravity is gravity (I'm sure you know this one).

Now that we have the net force, we can use Newton's second law (F = ma) to find the acceleration. Just divide the net force by the mass of m1, and voila! You've got your answer.

b) Alright, let's move on to the tension of the string. We know that the tension in the string is the same for both blocks because it's just that kind of equal opportunity string. And since m1 and m2 are connected by this magical (ahem, massless) string, their accelerations are also the same.

Using the equation for m2, we can find the net force acting on it. The only horizontal force acting on m2 is the tension in the string, while the vertical force is just the weight of m2. So, the net force is given by:

Net force = Tension - Weight of m2

We can find the weight of m2 using the same equation as before (mass × gravity), with the mass of m2 being 5.7 kg.

Once again, we can use Newton's second law (F = ma) to find the acceleration. And since the accelerations of both blocks are the same, you can use the acceleration you found in part (a) for this one as well.

Wow, that was quite a workout for our brain cells, wasn't it? I hope you found that helpful, or at the very least, slightly amusing.

To determine the acceleration of m1, we need to calculate the net force acting on it.

Given:
m1 = 25.5 kg
m2 = 5.7 kg
θ = 32.1 degrees
F = 239.3 N
μ = coefficient of kinetic friction = 0.225

First, let's calculate the force of friction acting on m1 using the coefficient of kinetic friction:

Frictional force = μ * m1 * g
= 0.225 * 25.5 kg * 9.8 m/s^2
≈ 56.10075 N

Next, let's determine the force component acting along the direction of acceleration of m1:

Force component along the direction of m1's acceleration = F * cos(θ)
= 239.3 N * cos(32.1 degrees)

Now, we can calculate the net force acting on m1:

Net force = Force component along the direction of acceleration - Frictional force
= F * cos(θ) - Frictional force

Now, let's calculate the tension in the string connecting m1 and m2:

Tension in the string = m2 * g - m2 * a

Since both m1 and m2 are connected by a massless string, the acceleration of m1 is also the acceleration of m2:

a = (Force component along the direction of acceleration - Frictional force) / (m1 + m2)

Substituting the given values into the above expressions will give us the final answers for both the acceleration of m1 and the tension in the string.

To determine the acceleration of m1, we need to analyze the forces acting on the system.

Let's consider the forces acting on m1:

1. The tension force (T) in the string, which pulls m1 to the right.
2. The force of gravity (mg1), which acts vertically downward.
3. The friction force (f) opposing the motion.

The force equation for m1 in the x-direction is:

T - f = m1 * a

Now let's analyze the forces acting on m2:

1. The force of gravity (mg2), which acts vertically downward.
2. The tension force (T) in the string, which pulls m2 to the left.

The force equation for m2 in the x-direction is:

T = m2 * a

To find the acceleration (a), we need to find the tension force (T) and the friction force (f).

To find the tension force (T), we can use the force equation for m2:

T = m2 * a

Substituting this value for T in the force equation for m1, we get:

m2 * a - f = m1 * a

Now, let's find the friction force (f). The friction force can be calculated using the coefficient of kinetic friction (μk) and the normal force (N). The normal force is equal to the force of gravity acting on m1:

N = mg1

The friction force can now be calculated using:

f = μk * N

Substituting the value of N, we get:

f = μk * mg1

Finally, substituting the values of T and f back into the equation:

m2 * a - μk * mg1 = m1 * a

Simplifying this equation, we can solve for acceleration, a:

a = (m2 * g - μk * m1 * g) / (m1 + m2)

Substituting the known values of m1, m2, g, and μk, we can calculate the acceleration, a.

For part b, to determine the tension in the string, we can use the equation for the tension force:

T = m2 * a

Substituting the value of a obtained from part a, we can calculate the tension of the string.