Question

In Fig. 6-24, a forceacts on a block weighing ${}^{\mathbf{45}\mathbf{}\mathbf{N}}$.The block is initially at rest on a plane inclined at angle ${}^{\mathbf{\theta }\mathbf{=}{\mathbf{15}}^{\mathbf{0}}}$to the horizontal. The positive direction of the x axis is up the plane. Between block and plane, the coefficient of static friction is ${}^{{\mathbf{\mu }}_{\mathbf{s}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{50}}$and the coefficient of kinetic friction is ${}^{{\mathbf{\mu }}_{\mathbf{k}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{34}}$. In unit-vector notation, what is the frictional force on the block from the plane when is (a) ${}^{\mathbf{(}\mathbf{-}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{N}\mathbf{)}\hat{\mathbf{i}}}$, (b) ${}^{\mathbf{(}\mathbf{-}\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{N}\mathbf{)}\hat{\mathbf{i}}}$, and (c) ${}^{\mathbf{(}\mathbf{-}\mathbf{15}\mathbf{}\mathbf{N}\mathbf{)}}$?

Short answer

(a) The static frictional force is equivalent to the ${}^{\mathbf{16}\mathbf{.}\mathbf{6}\mathbf{}\mathbf{N}}$force along $\hat{\mathrm{i}}$.

(b) The static frictional force is equivalent to the ${}^{\mathbf{19}\mathbf{.}\mathbf{6}\mathbf{}\mathbf{N}}$force along $\hat{\mathrm{i}}$.

(c) The frictional force or kinetic frictional force acting on the block is ${}^{\mathbf{14}\mathbf{.}\mathbf{78}\mathbf{}\mathbf{N}}$along.

Step-by-step solution

Given

Weight,${}^{\mathbf{W}\mathbf{=}\mathbf{45}\mathbf{}\mathbf{N}}$

Coefficient of static friction,${}^{{\mathbf{\mu }}_{\mathbf{s}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{50}}$

Coefficient of kinetic friction,${}^{{\mathbf{\mu }}_{\mathbf{k}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{34}}$

Inclined angle,${}^{\mathbf{\theta }\mathbf{=}{\mathbf{15}}^{\mathbf{0}}}$

Determining the concept

The problem is based on Newton’s second law of motion which states that the rate of change of momentum of a body is equal in both magnitude and direction of the force acting on it. Use the Newton's 2nd law of motion along vertical and horizontal direction. According to Newton's 2nd law of motion, a force applied to an object at rest causes it to accelerate in the direction of the force.

Formula:

$F_{net}=\sum ma$

where, F is the net force, m is mass and a is an acceleration.

Determining the free body diagram

Free body Diagram of sled over the inclined plane:

(a) Determining the frictional force on the block when P⏞U~=(5.0 N)(i)^

The maximum static frictional force, $f_{max}={\mu }_{s}N,$

By using Newton’s 2nd law of motion along vertical direction,

$N=W\cos \left({15}^0\right)$

Thus,${}^{f_{max}={\mu }_{s}N={\mu }_s(W\cos \left(15\right))=(0.50)\left(45\right)\cos \left({15}^0\right)=21.73N}$

Similarly, the maximum kinetic friction,

$$\begin{gathered} f_{max}={\mu }_{k}N \\ ={\mu }_k(W\cos \left(15\right)) \\ =(0.34)\left(45\right)\cos \left(15\right) \\ =14.78N \end{gathered}$$

If force P is applied along the inclination, then it’s effect on the frictional force by using Newton’s 2nd law along horizontal direction (x) is,

$\sum F_x=m{a}_x$

since, the block is not moving, ${}^{a_x=0}$

$$\begin{gathered} f_s-P-W\sin \left(\theta \right)=0 \\ {f}_s=P+W\sin \left(\theta \right) \end{gathered}$$

Since, ${}^{f_s}$is along the opposite direction of it should be ${}^{16.6\hat{\mathrm{i}}}$. This force is less than maximum static frictional force. So, the static frictional force is equivalent to the ${}^{16.6N}$force along $\hat{\mathrm{i}}$

(b) Determining the frictional force on the block when

Similarly, here also ${}^{f_s<f_{max}}$So, the static frictional force is equivalent to the 16.6N force alongrole="math" localid="1654156238627" $\hat{i}$.

(c) Determining the frictional force on the block when 

Since, ${}^{f_s>f_{max}}$, then the block begins to slide. Therefore, the kinetic frictional comes to play. So, the frictional force or kinetic frictional force acting on the block is role="math" localid="1654156555096" ${}^{14.78\mathrm{N}}$along the