Q102P - University Physics with Modern Physics

Question

You are riding in a school bus. As the bus rounds a flat curve at constant speed, a lunch box with mass 0.500 kg, suspended from the ceiling of the bus by a string 1.80 m long, is found to hang at rest relative to the bus when the string makes an angle of 30.0° with the vertical. In this position the lunch box is 50.0 m from the curve’s center of curvature. What is the speed v of the bus?

Step-by-Step Solution

Answer

The speed of the bus is 16.82 m/s.

Step 1: Identify the given data

• The mass of the lunch box, m = 0.500 kg .
• The angle, $\theta =30°$ .
• The distance of the lunch box from the curve’s center of curvature, R = 50 m .

Step 2: Concept/Significance of friction force

The force that acts between two objects in contact is known as friction force. If the object is moving, then there will be a kinetic friction force, and if the object is at rest, then there will be a static friction force.

Step 3: Find the speed v of the bus

Draw the free-body diagram of the lunch box.

 

 

The force acting on the lunch box along the x-axis is given by,

$$\begin{gathered} T \sin \theta =\underset{ }{\sum F_x} \\ T \sin \theta =m{a}_{rad} \\ T \sin \theta =\underset{ }{{\displaystyle \frac{m{v}^2}{R}}} .......\left(1\right) \end{gathered}$$ 

 

Here,  T is the tension force, and  v is velocity.

 

The force acting on the lunch box along the y-axis is given by,

 

$$\begin{gathered} \underset{ }{ \sum }{F}_y=0 \\ T-\cos \theta -mg=0 \\ T \cos \theta =mg ..........\left(2\right) \end{gathered}$$ 

 

Divide equation (1) by (2) as:

 

$$\begin{gathered} \tan \theta =\frac{v^2}{gR} \\ v^2=gR\tan \theta \\ v=\sqrt{gR\tan \theta } ........\left(3\right) \end{gathered}$$ 

 

Substitute $9.8 m/s^2$  for g , 50 m  for R , and  $30°$ for $\theta$ in equation (3), and we get,

 

$$\begin{gathered} v=\sqrt{\left(9.8 m/s^2\right)\times \left(50 m\right)\times \left(\tan \left(30°\right)\right)} \\ =16.82 m/s \end{gathered}$$  

 

Therefore, the speed of the bus is 16.82 m/s .