Two blocks (one of which is on a ramp) are attached via a string looped over a pulley. The block on the incline has a mass of 5.0 kg; the block hanging downward (suspended by the pulley) has a mass of 7.0 kg. Assume the usual things (non-stretchy massless string, frictionless lightweight pully, air resistance / friction is negligible, etc.) What angle of incline (measured against the horizontal) allows the blocks to have constant velocity?
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To determine the angle of incline that allows the blocks to have constant velocity, we need to consider the forces acting on the two blocks.
Let's break down the forces acting on each block:
1. Block on the incline:
- Weight (mg): The weight of the block acts vertically downward and can be split into two components:
- Perpendicular to the incline: mg * cos(θ), where θ is the angle of incline.
- Parallel to the incline: mg * sin(θ).
- Normal force (N): Acts perpendicular to the incline and cancels out the perpendicular component of the weight.
- Frictional force (f): Acts parallel to the incline and opposes the motion.
2. Hanging block:
- Weight (mg): The weight of the block acts vertically downward.
- Tension (T): The tension in the string acts upward and causes the block to accelerate or decelerate depending on its magnitude.
Since the blocks are connected by a string looped over a pulley, the tension in the string is the same throughout.
For the blocks to have constant velocity, the acceleration of both blocks must be zero. In other words, the net force acting on each block must be zero.
Next, we'll set up and solve the equations of motion for each block:
1. Block on the incline:
The net force acting parallel to the incline is given by:
Net force = T - f
The net force acting perpendicular to the incline is given by:
Net force = N - mg * cos(θ)
Since the block is not accelerating, the net force must be zero:
T - f = 0 ----(1)
N - mg * cos(θ) = 0 ----(2)
2. Hanging block:
The net force acting vertically is given by:
Net force = T - mg
Since the block is not accelerating vertically, the net force must be zero:
T - mg = 0 ----(3)
Solving these equations will give us the relationship between the tension T and the angle of incline θ.
Equation (3) gives:
T = mg ----(4)
Substituting equation (4) into equation (1):
mg - f = 0
f = mg ----(5)
Substituting equation (5) into equation (2):
N - mg * cos(θ) = 0
N = mg * cos(θ) ----(6)
Now, using the force diagram for the block on the incline, we see that the frictional force f is given by:
f = μN
where μ is the coefficient of friction. Since the problem statement says there is no friction, we have μ = 0.
Substituting μ = 0 into equation (5):
f = mg = 0
Now, substituting f = 0 into equation (1):
T - 0 = 0
T = 0
Since T = 0, equation (4) gives us:
mg = 0
This tells us that the tension in the string and the normal force cancel each other out, resulting in zero acceleration. Therefore, the blocks can have constant velocity for any angle of incline (θ) in this scenario.