# Box sliding down an inclined plane

• diegocas
In summary, a box is placed on an inclined plane and no friction exists between the box and the plane or between the plane and the floor. The masses of the box and inclined plane are given as m_1 and m_2, respectively. Equations for the motion of both the box and the inclined plane can be written using Newton's equations. By considering the motion of the box relative to the inclined plane, the acceleration can be determined. The final step is to solve the equations to find the unknowns.
diegocas
Homework Statement

A box is placed on an inclined plane of angle $$\theta$$. There is no fiction between the box and the plane nor between the inclined plane and the floor. The mass of the box is $$m_1$$ and the mass of the inclined plane is $$m_2$$.

Find the acceleration of the box and that of the inclined plane.

The attempt at a solution

It is easy to write down Newton's equations for both the box and the inclined plane.
Here they are:

$$N_1 \sin \theta = m_1 a_{1x}$$

$$N_1 \cos \theta - m_1 g = m_1 a_{1y}$$

$$- N_1 \sin \theta = m_2 a_{2x}$$

$$N_2 - N_1 \cos \theta - m_2 g = 0$$

where $$N_1$$ is the contact force between the box and the inclined plane and $$N_2$$ is the contact force between the inclined plane and the floor.

These are four equations in the unknowns $$N_1, N_2, a_{1x}, a_{1y}, a_{2x}$$.

How am I supposed to solve them?

My idea is the following: since the box is sliding down the inclined plane, the motion with respect to the inclined plane is easier. I mean the acceleration of the box relative to the inclined plane must be in the direction of the inclined plane. That is, the vector

$$\vec{a}_{box/plane} = \vec{a}_{box} - \vec{a}_{plane}$$

should be in the direction of the plane. Thus

$$\tan \theta = \frac{a_{box/plane,y}}{a_{box/palne,x}} = \frac{a_{box,y}-a_{plane,y}}{a_{box,x}-a_{plane,x}} = \frac{a_{1y}}{a_{1x}-a_{2x}}.$$

Is that right?

You are on the right track. Go ahead.

ehild

## 1. What is the formula for calculating the acceleration of a box sliding down an inclined plane?

The formula for calculating the acceleration of a box sliding down an inclined plane is a = gsinθ, where g is the acceleration due to gravity (9.8 m/s²) and θ is the angle of the incline.

## 2. How does the mass of the box affect its acceleration down the inclined plane?

The mass of the box does not affect its acceleration down the inclined plane. The acceleration is only dependent on the angle of the incline and the acceleration due to gravity.

## 3. What is the relationship between the angle of the incline and the acceleration of the box?

The acceleration of the box is directly proportional to the sine of the angle of the incline. This means that as the angle of the incline increases, the acceleration of the box also increases.

## 4. How does friction affect the motion of the box down the inclined plane?

Friction can act in two ways on a box sliding down an inclined plane. If the coefficient of friction is high, it can slow down the motion of the box. However, if the coefficient of friction is low, it can help the box slide down the inclined plane more smoothly.

## 5. What is the difference between kinetic and static friction in relation to a box sliding down an inclined plane?

Kinetic friction is the force that acts on an object while it is in motion, while static friction is the force that prevents an object from moving. In the case of a box sliding down an inclined plane, kinetic friction would act to slow down the box's motion, while static friction would act to keep the box from sliding down the incline too quickly.

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