A 2 kg mass resting on a smooth plane inclined 20 degrees to the horizontal. A cord which is parallel to the plane passes over a massless, frictionless pulley to a 4 kg mass which will drop vertically when released. What will be the speed of the 4 kg mass 4 seconds after it is released from rest?

on the plane, force down the plane mg*sinTheta. Force up the plane=4g

net force= 4g-2gSintheta=(6g)a
solve for a.

then, speed=at

To find the speed of the 4 kg mass after 4 seconds, we need to use the principles of motion and forces.

First, let's calculate the acceleration of the system. The acceleration is equal to the net force divided by the total mass.

The weight of the 2 kg mass can be split into two components: one perpendicular to the ramp and one parallel to the ramp. The component of the weight acting parallel to the ramp will cause the 4 kg mass to accelerate.

The force due to gravity acting parallel to the ramp can be calculated using the formula:
Force parallel = mass * acceleration due to gravity * sin(theta)

where mass is the mass of the 2 kg mass, acceleration due to gravity is approximately 9.8 m/s^2, and theta is the angle of the incline (20 degrees).

Force parallel = 2 kg * 9.8 m/s^2 * sin(20 degrees)
= 2 * 9.8 * 0.342
= 6.6864 N

Next, we can calculate the acceleration of the system using Newton's second law, which states that the acceleration is equal to the net force divided by the total mass.

Total mass = mass of 2 kg mass + mass of 4 kg mass
= 2 kg + 4 kg
= 6 kg

Acceleration = Force parallel / Total mass
= 6.6864 N / 6 kg
= 1.1144 m/s^2

Now that we have the acceleration, we can find the speed of the 4 kg mass after 4 seconds.

Using the equation of motion:
Final velocity = Initial velocity + (acceleration * time)

Since the 4 kg mass is released from rest, the initial velocity is 0 m/s.

Final velocity = 0 + (1.1144 m/s^2 * 4 s)
= 4.4576 m/s

Therefore, the speed of the 4 kg mass 4 seconds after being released from rest is approximately 4.4576 m/s.

To find the speed of the 4 kg mass after 4 seconds, we need to analyze the forces acting on the system. Here are the steps to solve the problem:

1. Draw a free body diagram: Visualize the forces acting on each object.

- For the 2 kg mass on the inclined plane:
- The weight of the mass (mg) acts vertically downwards.
- The normal force (N) acts perpendicular to the inclined plane.
- The tension in the cord (T) acts parallel to the inclined plane.

- For the 4 kg mass:
- The weight of the mass (mg) acts vertically downwards.
- The tension in the cord (T) acts upwards, as it is connected to the 2 kg mass.

2. Resolve the forces: Split each force into components parallel and perpendicular to the inclined plane.

- For the 2 kg mass:
- The weight force (mg) can be split into a component parallel to the plane (mg*sin(20°)) and a component perpendicular to the plane (mg*cos(20°)).
- The normal force (N) acts perpendicular to the plane and has no component parallel to it.
- The tension force (T) also has no component perpendicular to the plane.

- For the 4 kg mass:
- The weight force (mg) acts vertically downwards and has no component parallel or perpendicular to the inclined plane.
- The tension force (T) acts upwards and has no component parallel or perpendicular to the inclined plane.

3. Apply Newton's second law: Write the equations of motion for each object along the directions of acceleration. Since we are interested in the speed, we will work with the magnitude of the acceleration.

- For the 2 kg mass: The acceleration a₁ can be calculated from the component of the weight force parallel to the plane (mg*sin(20°)) and the tension force.
Let T = tension and a₁ = acceleration of 2 kg mass.
T - mg*sin(20°) = m₁ * a₁

- For the 4 kg mass: The acceleration a₂ can be calculated from the tension force.
Let T = tension and a₂ = acceleration of 4 kg mass.
mg - T = m₂ * a₂

4. Solve for the acceleration: Use the equations obtained in step 3 to calculate the acceleration of both masses.

- For the 2 kg mass:
T - mg*sin(20°) = m₁ * a₁

- For the 4 kg mass:
mg - T = m₂ * a₂

Since the masses are connected and will move together, they will have the same acceleration a.

Combining the two equations:

T - mg*sin(20°) = m₁ * a
mg - T = m₂ * a

Substitute the given values:
m₁ = 2 kg, m₂ = 4 kg, g = 9.8 m/s²

2a - 2 * 9.8 * sin(20°) = 2a
19.6 * sin(20°) = 2a
a ≈ 1.128 m/s²

5. Calculate the distance traveled: Use the kinematic equation to find the distance covered by the 4 kg mass in 4 seconds.

Since the mass starts from rest, the initial velocity u₁ = 0.

The distance d can be calculated using the equation:
d = u₁ * t + (1/2) * a * t²

Using u₁ = 0, a = 1.128 m/s², and t = 4 s:
d = 0 + (1/2) * 1.128 * (4)²
d ≈ 9.024 m

6. Find the final velocity: Calculate the final velocity v₂ using the equation:
v₂ = u₂ + a * t

Since the 4 kg mass starts from rest, u₂ = 0.

Using a = 1.128 m/s² and t = 4 s:
v₂ = 0 + 1.128 * 4
v₂ ≈ 4.512 m/s

Therefore, the speed of the 4 kg mass 4 seconds after it is released from rest is approximately 4.512 m/s.