A block of mass m1 = 26 kg rests on a wedge of angle θ = 48∘ which is itself attached to a table (the wedge does not move in this problem). An inextensible string is attached to m1, passes over a frictionless pulley at the top of the wedge, and is then attached to another block of mass m2 = 7 kg. The coefficient of kinetic friction between block 1 and the plane is μ = 0.5. The string and wedge are long enough to ensure neither block hits the pulley or the table in this problem, and you may assume that block 1 never reaches the table. take g to be 9.81 m/s2
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To get the answer to this problem, we need to determine the acceleration of the system and the tension in the string.
Let's break down the problem into different components:
1. Forces on block 1:
- The weight of block 1, W1 = m1 * g, where g is the acceleration due to gravity.
- The normal force, N1, which is perpendicular to the incline.
- The force of friction, Ff1, which acts parallel to the incline and opposes motion.
2. Forces on block 2:
- The weight of block 2, W2 = m2 * g.
3. Forces on the wedge:
- The normal force, Nw, which acts perpendicular to the incline.
- The weight of the wedge, Ww = 0, as the wedge is assumed to be massless.
Now, we need to resolve the forces into components parallel and perpendicular to the incline:
1. Forces on block 1 along the incline:
- The weight component along the incline, W1_parallel = m1 * g * sin(θ).
- The normal force component along the incline, N1_parallel = N1 * cos(θ).
- The force of friction component along the incline, Ff1_parallel = μ * N1_parallel, where μ is the coefficient of friction.
2. Forces on block 1 perpendicular to the incline:
- The weight component perpendicular to the incline, W1_perpendicular = m1 * g * cos(θ).
- The normal force component perpendicular to the incline, N1_perpendicular = N1 * sin(θ).
Using the equations above, we can now analyze the motion of the blocks.
Since there is an inextensible string connecting the blocks, they have the same acceleration. Let's call this acceleration "a" and assume it moves down the incline.
For block 1, the net force acting along the incline is given by:
F_net1 = W1_parallel - Ff1_parallel.
Using Newton's second law (F_net1 = m1 * a), we can solve for the acceleration:
m1 * a = m1 * g * sin(θ) - μ * N1 * cos(θ).
Next, let's analyze the forces on block 2:
For block 2, the only force acting on it is its weight, W2 = m2 * g.
Using Newton's second law (F_net2 = m2 * a), we can solve for the acceleration:
m2 * g = m2 * a.
Now, we need to relate the acceleration "a" of the blocks to the angle θ and the system's parameters.
Since the blocks are connected by an inextensible string, their accelerations are related:
a = a1 = a2.
Considering the geometry of the system, we can relate the accelerations by:
a = a1 = a2 = (g * sin(θ)) / (1 + (m1/m2)).
Finally, we can find the tension in the string:
The net force acting on block 1 is given by:
F_net1 = W1_parallel - Ff1_parallel.
Using Newton's second law (F_net1 = m1 * a), we can solve for the tension in the string:
T = m1 * a + Ff1_parallel.
Substituting the values of "a" and Ff1_parallel obtained earlier, we can calculate the tension T.
Therefore, to find the value of the acceleration "a" and the tension in the string, we need to substitute the given values of m1, m2, θ, μ, and g into these equations and solve them step by step.