Question

Consider an assembly of two blocks, where block 1 slides on an inclined plane and attached to a block 2 via a pulley anchored on the top edge of the plane allowing it to move vertically.

In this problem, the fact that the length of the string does not change imposes a constraint on relative accelerations of the two blocks.
Find a relationship between the x component of the acceleration of block 2, a2x, and the acceleration of block 1, a1x. Pay careful attention to signs.

Answer 1

a = g (m₂ -m₁ sin θ) / (m₁ + m₂)

Step-by-step explanation:

To solve this we must locate a reference system in the inclined plane, in this case the axis eg. it is parallel to the plane and the axis and is perpendicular to the plane, we take the direction to the right as positive.

Let's write Newton's second law for block 1 on the plane

X axis

T - Wₓ = m₁ a

Y Axis

N –
`W_(y)` = 0

We use trigonometry to find the components of the weight

sin θ = Wₓ / W

Wₓ = W sin θ

cos θ = W_{y} / W

W_{y}= W cos θ

We substitute

T - m₁ g sin θ = m₁ a

N = m₁ g cos θ

We write Newton's equations for block 2 that is hanging.

Note that if block 1 goes up, block 2 must go down, therefore for this block the positive direction is down.

W₂ - T = m₂ a

Let's write the system of equations

T - m₁ g sin θ = m₁ a

m₂ g - T = m₂ a

Let's add

m₂ g - m₁ g sin θ = (m₂ + m₁) a

a = g (m₂ -m₁ sin θ) / (m₁ + m₂)

Acceleration is the same for both blocks as they are connected by a rope