Problem 2 - University Physics with Modern Physics

Question

Cavendish Experiment. In the Cavendish balance apparatus shown in Fig. \(13.4,\) suppose that \(m_{1}=1.10 \mathrm{kg}, m_{2}=\) \(25.0 \mathrm{kg},\) and the rod connecting the \(m_{1}\) pairs is 30.0 \(\mathrm{cm}\) long. If, in each pair, \(m_{1}\) and \(m_{2}\) are 12.0 \(\mathrm{cm}\) apart center to center, find (a) the net force and (b) the net torque (about the rotation axis) on the rotating part of the apparatus. (c) Does it seem that the torque in part (b) would be enough to easily rotate the rod? Suggest some ways to improve the sensitivity of this experiment.

Step-by-Step Solution

Answer

(a) Net force is \( 2.54 \times 10^{-7} \text{ N} \), (b) net torque is \( 3.82 \times 10^{-8} \text{ Nm} \), (c) torque may not suffice; improve sensitivity by using longer arms or larger masses.

Step 1: Understand the Problem

In the Cavendish experiment, we need to calculate the gravitational force between two masses and then find the torque produced by this force. The masses are given as \( m_1 = 1.10 \) kg and \( m_2 = 25.0 \) kg, and they are 12.0 cm apart. The rod connecting the \( m_1 \) pairs is 30.0 cm long.

Step 2: Calculate the Gravitational Force

The gravitational force between two masses \( m_1 \) and \( m_2 \) that are \( r \) meters apart can be calculated using the formula: \[ F = \frac{{G imes m_1 imes m_2}}{{r^2}} \]where \( G = 6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2 \). Convert the distance to meters \( r = 0.12 \) m. Substitute the values to find \( F \).\[ F = \frac{{6.674 \times 10^{-11} \times 1.10 \times 25}}{(0.12)^2} \approx 1.27 \times 10^{-7} \text{ N} \]

Step 3: Calculate Net Force

Since there are two pairs of \( m_1 \) and \( m_2 \), the net gravitational force on the entire system is just twice the force calculated between one pair: \[ F_{net} = 2 \times 1.27 \times 10^{-7} = 2.54 \times 10^{-7} \text{ N} \]

Step 4: Calculate Torque Due to Gravitational Force

Torque \( \tau \) can be calculated using the formula: \[ \tau = r_{lever} \times F \times \sin(\theta) \]where \( r_{lever} = 0.15 \) m is the distance from the axis of rotation to the line of action of the force and \( \theta = 90^\circ \) (since the force is horizontal), making \( \sin(90^\circ) = 1 \). Therefore, \[ \tau = 0.15 \times 1.27 \times 10^{-7} = 1.91 \times 10^{-8} \text{ Nm} \]For two pairs, the net torque is:\[ \tau_{net} = 2 \times 1.91 \times 10^{-8} = 3.82 \times 10^{-8} \text{ Nm} \]

Step 5: Evaluate the Rotation of Rod

The net torque calculated indicates that the force is very small, which might not be sufficient to easily rotate the rod in the experiment given the small magnitude.

Step 6: Suggestions for Improving Sensitivity

To improve the sensitivity of this experiment, consider using a longer arm or increasing the mass of \( m_1 \) to make the torque larger. Additionally, using a more sensitive apparatus to measure smaller torques or reducing any friction or air resistance might improve the sensitivity.

Key Concepts

Gravitational Force Calculation

The Cavendish experiment is a fascinating demonstration of gravitational attraction between masses. To calculate the gravitational force in this experiment, you use the formula:\[ F = \frac{G \times m_1 \times m_2}{r^2} \]where:

• \( G = 6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2 \) is the gravitational constant,
• \( m_1 = 1.10 \) kg and \( m_2 = 25.0 \) kg are the masses,
• \( r = 0.12 \) meters is the distance between the centers of the two masses.

By converting the given distance from centimeters to meters, you ensure the formula uses consistent units. Plug in the values to find the gravitational force:\[ F = \frac{6.674 \times 10^{-11} \times 1.10 \times 25}{(0.12)^2} \approx 1.27 \times 10^{-7} \text{ N} \]For the entire system with two pairs of masses, the net gravitational force is twice this calculated force:\[ F_{net} = 2 \times 1.27 \times 10^{-7} = 2.54 \times 10^{-7} \text{ N} \]This calculated force helps in understanding the minute interactions of gravity at play, emphasizing why the Cavendish experiment is crucial in determining the gravitational constant.

Torque Calculation

When considering the Cavendish experiment, the concept of torque is essential since it describes how forces cause objects to rotate. Torque \( \tau \) is calculated using:\[ \tau = r_{lever} \times F \times \sin(\theta) \]Here, \( r_{lever} = 0.15 \) meters is the lever arm distance from the axis of rotation to the force's line of action. The angle \( \theta = 90^\circ \) (with force applied horizontally) makes \( \sin(90^\circ) = 1 \), simplifying the equation to:\[ \tau = 0.15 \times 1.27 \times 10^{-7} = 1.91 \times 10^{-8} \text{ Nm} \]Since there are two pairs of interacting masses, the total net torque becomes:\[ \tau_{net} = 2 \times 1.91 \times 10^{-8} = 3.82 \times 10^{-8} \text{ Nm} \]The calculated torque reflects the experiment's delicate balance. Its small value underscores the precision needed in measurements and interactions when observing gravitational effects between masses.

Sensitivity Improvement Techniques

Enhancing the sensitivity of the Cavendish experiment is critical for more accurate gravitational force measurements. Here are some techniques to improve sensitivity:

• Lengthen the Lever Arm: By extending the rod connecting the masses, you increase the lever arm \( r_{lever} \), which amplifies the torque, making small forces more discernible.


• Increase Mass: Using larger masses can enhance the gravitational pull and subsequent torque, aiding in more notable rotational effects.


• Reduce Friction: Ensuring that the apparatus operates with minimal friction and air resistance allows for easier observation of subtle torque effects. This might involve using smoother supports or operating in a vacuum environment.


• Use Sensitive Measuring Devices: Implementing high-precision devices, such as laser interferometers, to detect minute changes in angular positions can considerably enhance sensitivity.

Employing these adjustments not only improves the experiment's efficacy but also helps refine our understanding and measurements of gravitational forces within controlled settings.