Q9E - University Physics with Modern Physics

Question

A 8.00-kg ball is tied to the end of a string 1.60 m long and swung in a vertical circle. (a) During one complete circle, starting anywhere, calculate the total work done on the ball by (i) the tension in the string and (ii) gravity. (b) Repeat part (a) for motion along the semicircle from the lowest to the highest point on the path.

Step-by-Step Solution

Answer

1. 0 J and 0 J
2. 0 J and -25.1 J

Step 1: Identification of the given data

The given data is listed below as-

• The mass of the ball is, $m=0.8 \mathrm{kg}$ 
• The length of a string, $l=1.6 \mathrm{m}$ 

Step 2: Significance of the work done

The Work done along a path P₁ to P₂ is given by

$\mathbf{W}\mathbf{=}{\mathbf{\int }}_{{\mathbf{P}}_{\mathbf{1}}}^{{\mathbf{P}}_{\mathbf{2}}}\overrightarrow{\mathbf{F}}\mathbf{d}\overrightarrow{\mathbf{s}}$ 

The force of tension is perpendicular to the direction of motion if the motion of a ball is a perfect circle. Therefore, the work done by it is 0.

Step 3: Determination of work done on theball during one complete circle (a)

The force of gravity that is directed upwards is given by

${\overrightarrow{F}}_g=-mg\hat{z}$ 

The work done on the ball is given by 

 $W_g=-mg{\int }_{P_1}^{P_2}dz=-mg\left(z_2-z_1\right)$

When the ball went around in a full circle, $W_g=0$

The work done by Tension is zero.

Thus, the magnitude of work done on the ball by tension is zero and when the ball went around a full circle is also 0.

Step 4: Determination of work done on the ball for motion along the semicircle from the lowest to the highest point on the path (b)

When the ball went from the lowest to the highest point, 

$W_g=-mg\left[l-\left(-l\right)\right]$ 

Here, m is the mass of the ball, g is the gravitational constant and l is the length of the spring.

 For $m=0.8 kg$, $l=1.6 m$, and $g=9.8 m{s}^{-2}$ 

$$\begin{gathered} W_g=-2mgl \\ =-2\times 0.8\times 9.8\times 1.6 J \\ =-25.1 J \end{gathered}$$

 

The work done by Tension is zero.

Thus, the magnitude of work done on the ball by tension is zero and when the ball went from the lowest to the highest point is -25.1 J.