A block of mass m1 = 3.08 kg on a friction less plane inclined at angle θ = 32.3° is connected by a cord over a massless, frictionless pulley to a second block of mass m2 = 2.51 kg hanging vertically (see the figure). (a) What is the acceleration of the hanging block (choose the positive direction down)? (b) What is the tension in the cord?

Question

A block of mass m1 = 3.08 kg on a friction less plane inclined at angle θ = 32.3° is connected by a cord over a massless, frictionless pulley to a second block of mass m2 = 2.51 kg hanging vertically (see the figure). (a) What is the acceleration of the hanging block (choose the positive direction down)? (b) What is the tension in the cord?

Answer 1

F = 2.51 (9.81) - 3.08 (9.81) sin 32.3

= 8.48 N

a = F/m = 8.48/(2.51+3.08)

2.51(9.81) - T = 2.51 a

Answer 2

To determine the acceleration of the hanging block and the tension in the cord, we can use the principles of Newton's second law and the free-body diagram.

Let's break down the problem step by step:

(a) To find the acceleration of the hanging block, we'll start by analyzing the forces acting on each block.

For the block on the inclined plane (m1), the force components are:

- Gravitational force: mg * sin(θ) acting down the incline.
- Normal force: N acting perpendicular to the incline.
- Tension in the cord: T acting parallel to the incline, upward.

For the hanging block (m2), the force components are:

- Gravitational force: mg acting downward.
- Tension in the cord: T acting upward.

Now, we can write the equations of motion for each block:

For m1:
m1 * a = mg * sin(θ) - T

For m2:
m2 * a = T - mg

Since both blocks share the same acceleration, we can equate the two equations:
m1 * a = mg * sin(θ) - T = T - m2 * g

Now, we can solve for the acceleration (a):

m1 * a + T = T - m2 * g

Simplify the equation:

m1 * a = T - T - m2 * g

m1 * a = -m2 * g

Finally, solve for the acceleration (a):

a = (-m2 * g) / m1

Plug in the given values:

m1 = 3.08 kg
m2 = 2.51 kg
g = 9.8 m/s²
θ = 32.3°

Calculate:

a = (-2.51 kg * 9.8 m/s²) / 3.08 kg

(a) The acceleration of the hanging block is approximately -8.01 m/s² (negative direction down).

(b) To find the tension in the cord, we can substitute the acceleration (a) into either of the previous equations and solve for T.

Using the equation for block m1:
m1 * a = mg * sin(θ) - T

Substitute the values:
(3.08 kg) * (-8.01 m/s²) = (3.08 kg) * (9.8 m/s²) * sin(32.3°) - T

Now, solve for T:

T = (3.08 kg) * (9.8 m/s²) * sin(32.3°) - (3.08 kg) * (-8.01 m/s²)

(b) The tension in the cord is approximately 35.72 N.