Question

In Fig. 6-45, a$\mathbf{1}\mathbf{.}\mathbf{34}\mathbf{}\mathbf{kg}$ball is connected by means of two massless strings, each of length$\mathbf{L}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{70}\mathbf{}\mathbf{m}$, to a vertical, rotating rod. The strings are tied to the rod with separation$\mathbf{d}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{70}\mathbf{}\mathbf{m}$and are taut. The tension in the upper string is$\mathbf{35}\mathbf{N}$. What are the

(a) tension in the lower string,

(b) magnitude of the net force${\overrightarrow{\mathbf{F}}}_{\mathbf{net}}$on the ball, and

(c) speed of the ball?

(d) What is the direction of$\mathbf{F}$?

Short answer

1. ${\mathrm{T}}_{\mathrm{l}}\text{\hspace{0.17em}is\hspace{0.17em}}8.74\text{\hspace{0.17em}N}$.
2. ${\mathrm{F}}_{\mathrm{net},\mathrm{st}}\text{is}37.9\text{\hspace{0.17em}N}$.
3. $\mathrm{v}\text{\hspace{0.17em}is\hspace{0.17em}}6.45\text{\hspace{0.17em}m/s}$.
4. The direction of ${\overrightarrow{\mathrm{F}}}_{\mathrm{net},\mathrm{str}}$is leftward.

Step-by-step solution

Given data

• Mass of ball,$\mathrm{m}=1.34\text{\hspace{0.17em}kg}$.
• Length of each string,$\mathrm{L}=1.70\text{\hspace{0.17em}m}$.
• The separation between two strings tied to rod,$\mathrm{d}=1.70\text{\hspace{0.17em}m}$.
• Tension in the upper string,${\mathrm{T}}_{\mathrm{u}}=35\text{\hspace{0.17em}N}$ .

To understand the concept

Using the concept of centripetal force and applying Newton's second law, we can solve the given problem. Note that the tensions in the strings provide the source of centripetal force.

Draw the free body diagram and write the force equations

The given system consists of a ball connected by two strings to a rotating rod. The tensions in the strings provide the source of centripetal force.

The free-body diagram for the ball is shown below. ${\overrightarrow{\mathrm{T}}}_{\mathrm{u}}$is the tension exerted by the upper string on the ball, ${\overrightarrow{\mathrm{T}}}_{\mathrm{l}}$is the tension in the lower string, and role="math" localid="1661234901128" $m$is the mass of the ball. Note that the tension in the upper string is greater than the tension in the lower string. It must balance the downward pull of gravity and the force of the lower string.

We take the$+\mathrm{x}$direction to be leftward (toward the center of the circular orbit) and$+\mathrm{y}$upward. Since the magnitude of the acceleration is$\mathrm{a}={\mathrm{v}}^2/\mathrm{R}$, the$\mathrm{x}$component of Newton's second law is,

${\mathrm{T}}_{\mathrm{u}}\text{cos}\mathrm{\theta }+{\mathrm{T}}_{\mathrm{l}}\text{cos}\mathrm{\theta }=\frac{{\mathrm{mv}}^2}{\mathrm{R}}$

Where$\mathrm{v}$is the speed of the ball, and$\mathrm{R}$ is the radius of its orbit.
The$\mathrm{y}$ component is,

${\mathrm{T}}_{\mathrm{u}}\text{sin}\mathrm{\theta }-{\mathrm{T}}_{\mathrm{l}}\text{sin}\mathrm{\theta }-\mathrm{mg}=0$

The second equation gives the tension in the lower string:

${\mathrm{T}}_{\mathrm{l}}={\mathrm{T}}_{\mathrm{u}}-\mathrm{mg}/\text{sin}\mathrm{\theta }$.

(a) Calculate the tension in the lower string

Since the triangle is equilateral, the angle is$\mathrm{\theta }={30.0}^{\text{o}}$.

Thus,

${\mathrm{T}}_{\mathrm{l}}={\mathrm{T}}_{\mathrm{u}}-\frac{\mathrm{mg}}{\text{sin}\mathrm{\theta }}$

Substitute the values, and we get,

$\begin{array}{l}{\mathrm{T}}_{\mathrm{l}}=35.0\text{\hspace{0.17em}N}-\frac{\left(1.34\text{\hspace{0.17em}kg}\right)(9.80{\text{\hspace{0.17em}m/s}}^{\text{2}})}{\text{sin}{30.0}^{\circ }}\\ {\mathrm{T}}_{\mathrm{l}}=8.74\text{\hspace{0.17em}N}\end{array}$

Thus, ${\mathrm{T}}_{\mathrm{l}}\text{\hspace{0.17em}is\hspace{0.17em}}8.74\text{\hspace{0.17em}N}$.

(b) Calculate the magnitude of the net force F¯net on the ball 

The net force in the$\mathrm{y}$ direction is zero. In the$x$-direction, the net force has magnitude as:

${\mathrm{F}}_{\mathrm{net},\mathrm{st}}=({\mathrm{T}}_{\mathrm{u}}+{\mathrm{T}}_{\mathrm{l}})\text{cos}\mathrm{\theta }$

Substitute the values, and we get,

$\begin{array}{l}{\mathrm{F}}_{\mathrm{net},\mathrm{st}}=(35.0\text{\hspace{0.17em}N}+8.74\text{\hspace{0.17em}N})\text{cos}{30.0}^{\text{o}}\\ {\mathrm{F}}_{\mathrm{net},\mathrm{st}}=37.9\text{\hspace{0.17em}N}\end{array}$

Thus, ${\mathrm{F}}_{\mathrm{net},\mathrm{st}}\text{is}37.9\text{\hspace{0.17em}N}$.

(c) Calculate the speed of the ball 

The radius of the path is,

$\mathrm{R}=\mathrm{L}\text{cos}\mathrm{\theta }$

Substitute the values, and we get,

$\begin{array}{c}\mathrm{R}=\left(1.70\text{\hspace{0.17em}m}\right)\text{cos}{30}^{\text{o}}\\ =1.47\text{\hspace{0.17em}m}\end{array}$

Using this${\mathrm{F}}_{\mathrm{net},\mathrm{str}}={\mathrm{mv}}^2/\mathrm{R}$, we find the speed of the ball to be,

$\mathrm{v}=\sqrt{\frac{{\mathrm{RF}}_{\mathrm{net},\mathrm{str}}}{\mathrm{m}}}$

Substitute the values, and we get,

$\begin{array}{l}\mathrm{v}=\sqrt{\frac{(1.47\text{m})(37.9\text{N})}{1.34\text{\hspace{0.17em}kg}}}\\ \mathrm{v}=6.4\text{5\hspace{0.17em}m/s}\end{array}$

Thus, $\mathrm{v}\text{\hspace{0.17em}is\hspace{0.17em}}6.45\text{\hspace{0.17em}m/s}$.

(d) Calculate the direction of F 

The direction of ${\overrightarrow{\mathrm{F}}}_{\mathrm{net},\mathrm{str}}$is leftward (radially inward).