Question

The box shown on the rough ramp above is sliding up the ramp. calculate the acceleration of the box

Answer 1

We are given that a block is sliding up an incline. A diagram of the situation is given as follows:

To determine the acceleration we will add the forces parallel to the ramp, we will call this direction the x-direction:

`\Sigma F_x=-mg_x-F_f`

Where:

`\begin{gathered} m=\text{ mass} \\ g=\text{ acceleration of gravity} \\ mg_{}=\text{ weight} \\ mg_x=\text{x-component of the} \\ F_f=\text{ force of friction} \end{gathered}`

Now we determine the x-component of the weight by using the trigonometric function sine:

`\sin 40=(mg_x)/(mg)`

Now we multiply both sides by "mg":

`mg\sin 40=mg_x`

Now we substitute this value in the sum of forces:

`\Sigma F_x=-mg_{}\sin 40-F_f`

Now, to determine the force of friction we will use the following formula:

`F_f=\mu N`

Where:

`N=\text{ normal force}`

To determine the normal force we add the forces in the direction perpendicular to the ramp, we will call this direction the y-direction:

`\Sigma F_y=N-mg_y`

Where:

`mg_y=y-\text{component of the weight}`

Now, since there is no movement in the y-direction, the sum of forces is equal to zero:

`N-mg_y=0`

Now we solve for the normal force:

`N=mg_y`

Now we calculate the y-component of the weight using the trigonometric function cosine:

`N=mg\cos 40`

Now we substitute this value in the expression for the friction force:

`F_f=\mu mg\cos 40`

Now we substitute this value in the sum of forces in the x-direction:

`\Sigma F_x=-mg_{}\sin 40-\mu mg\cos 40`

Now, since the sum of forces is equivalent to the product of the mass by the acceleration we have:

`-mg_{}\sin 40-\mu mg\cos 40=ma`

We can take "-mg" as a common factor on the left side:

`-mg(_{}\sin 40+\mu\cos 40)=ma`

We can cancel out the mass:

`-g(_{}\sin 40+\mu\cos 40)=a`

Now we substitue the values:

`-(9.8(m)/(s^2))(\sin 40+0.2\cos 40)=a`

Now we solve the operations:

`-7.8(m)/(s^2)=a`

Therefore, the acceleration is -7.8 meters per second squared.