Chapter 4, Problem 10Q

To determine

Acceleration, when a force is applied, is at an angle in absence and presence of friction

Answer to Problem 10Q

Solution:

The acceleration decreases.

Explanation of Solution

a)In absence of friction

As it is clear from above figure that when a constant force F is applied on the box, which placed on a frictionless table, with the help of a rope, we can determine the magnitude of force F, which is responsible for creating acceleration a in the box of mass m, which can be given by the equation from Newton’s second law

$\begin{array}{l}F=ma\\ a=F/m \mathrm{\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots }(1)\end{array}$

Now for the same magnitude of force F is applied at an angle θ to the horizontal, we will get that force in vertical and horizontal direction by resolving that force. Magnitude of acceleration in horizontal direction decreases, as the same force F get resolve along horizontal and vertical direction as shown in resolving the force F along horizontal and vertical direction, we get,

$F_x=Fcos\theta$ in horizontal

$F_y=Fsin\theta$ in vertical

Only the horizontal component is responsible for pulling the block along horizontal direction while the vertical component of force is balanced due to the weight of the box. Therefore, the box moves due to force  $F_x$,corresponding acceleration is given as

$\begin{array}{l}F=ma\\ a=F\cos \theta /m \mathrm{\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots }(2)\end{array}$

Thus, acceleration decreases when the box is pulled horizontally at an angle

b)Considering friction

If we pull box horizontal, the frictional force along the X axis is

$F_x$= µ F

Where µ is coefficient of friction.

Then acceleration will be of

$\begin{array}{l}F=ma\\ a=\mu F/m \end{array}$

Horizontal forces when we pull at certain angle θ, then force will be $F_x$=µ $\cos \theta$

$\begin{array}{l}F=ma\\ a=\mu F\cos \theta /m\end{array}$

Thus, acceleration decreases when the box is pulled horizontally at an angle