The pulley is light and frictionless. Find the mass M1, given that M2 (7.00 kg) is moving downwards and accelerates downwards at 3.01 m/s2, that θ is 15.0°, and that μk is 0.270.

without a diagram, this makes no sense.

To find the mass M1, we can start by analyzing the forces acting on the system. Let's break it down step by step:

1. Draw a free-body diagram: Draw a diagram of the situation and label the forces acting on each object. In this case, we have M2 moving downwards, so the forces acting on it are its weight (mg) pointing downward and the tension (T) in the string pointing upwards.

2. Write down the equations of motion: Using Newton's second law, we can write down the equations of motion for each object. For M2, the equation is:
ΣF = m2 * a2
T - m2 * g = m2 * a2

3. Decompose the forces: Resolve the forces into their components along the vertical and horizontal axis. In this problem, the only vertical component is the weight, which is given by:
Fvertical = mg

4. Consider the friction force: The friction force acting on M1 can be determined using the equation:
Ffriction = μk * N
Here, N is the normal force, which is equal to the weight of M1.

5. Resolve the tension force: Since the pulley is light and frictionless, the tension force in the string is the same for both M1 and M2. We can use trigonometry to split it into horizontal and vertical components:
T = Tvertical / sin(θ)
Thorizontal = T * cos(θ)

6. Solve the equations: Now we can substitute the forces into the equations of motion for M2 and solve for M1. Since T is the same for both M1 and M2, we can substitute the expression for T into the equation for M2:
T - m2 * g = m2 * a2

Substitute the expression for T:
(Tvertical / sin(θ)) - m2 * g = m2 * a2

Substitute Fvertical:
(mg / sin(θ)) - m2 * g = m2 * a2

Substitute M2's values (m2 = 7.00 kg and a2 = 3.01 m/s^2):
(mg / sin(θ)) - (7.00 kg) * g = (7.00 kg) * (3.01 m/s^2)

Now, solve this equation for M1.

Note: It is important to double-check your calculations and units to ensure accuracy.