Final answer:
The mass of block 1 (m₁) required to keep block 2 moving up the incline at constant speed can be determined by balancing the forces along the incline, recognizing that since block 2's speed is constant, the tension in the rope (equal to the gravitational force on m₁) must equal the sum of the gravitational component down the incline and the kinetic frictional force for block 2. The mass of block 1 is found using the relationship m₁ = m₂ sin(θ) + μ m₂ cos(θ), and by substituting the given values into this equation.
Step-by-step explanation:
To find the mass of block 1 (m₁), we need to consider the forces acting on block 2 as it moves up the incline at constant speed, which implies that the net force along the incline is zero. We have gravity, normal force, frictional force, and the tension in the rope as forces of interest. Since block 2 is moving at constant speed, the net force on it is zero. The forces acting on block 2 are:
- The component of gravitational force parallel to the incline: m₂g sin(\theta)
- The frictional force opposing the motion: \mu m₂g cos(\theta)
Since it's in equilibrium, the tension (T) in the rope pulling it uphill must balance these two forces. Calculating T yields:
T = m₂g sin(\theta) + \mu m₂g cos(\theta)
This tension is also the force exerted by gravity on block 1, so we have:
T = m₁g
Equating the two expressions for the tension, we can solve for m₁:
m₁ = m₂ sin(\theta) + \mu m₂ cos(\theta)
Plugging in the given values:
m₁ = (10.1 kg * sin(29.5°)) + (0.240 * 10.1 kg * cos(29.5°))
To get the numerical value of m₁, you can use a calculator to compute the trigonometric functions and multiply by the given mass and coefficient of friction.