Problem 54 - University Physics with Modern Physics

Question

A 1200 -kg station wagon is moving along a straight highway at 12.0 \(\mathrm{m} / \mathrm{s}\) . Another car, with mass 1800 \(\mathrm{kg}\) and speed \(20.0 \mathrm{m} / \mathrm{s},\) has its center of mass 40.0 \(\mathrm{m}\) ahead of the center of mass of the station wagon (Fig. E8.54). (a) Find the position of the center of mass of the system consisting of the two automobiles. (b) Find the magnitude of the total momentum of the system from the given data. (c) Find the speed of the center of mass of the system. (d) Find the total momentum of the system, using the speed of the center of mass. Compare your result with that of part (b).

Step-by-Step Solution

Answer

(a) 24 m, (b) 50400 kg m/s, (c) 16.8 m/s, (d) matches 50400 kg m/s.

Step 1: Determine the Position of the Center of Mass

The position of the center of mass (CM) of a two-object system can be found using the formula:\[ x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]Where:- \(m_1 = 1200\, \mathrm{kg}\) is the mass of the station wagon,- \(x_1 = 0\, \mathrm{m}\) is the position of the station wagon (reference point),- \(m_2 = 1800\, \mathrm{kg}\) is the mass of the second car,- \(x_2 = 40\, \mathrm{m}\) is the position of the second car.Substitute the known values:\[ x_{\text{cm}} = \frac{(1200 \times 0) + (1800 \times 40)}{1200 + 1800} = \frac{0 + 72000}{3000} = \frac{72000}{3000} = 24\, \mathrm{m} \]Thus, the center of mass is 24 m ahead of the station wagon's position.

Step 2: Calculate the Total Momentum of the System

The momentum of a system is calculated by summing the momentum of its components. The momentum \(p\) of an object is the product of its mass \(m\) and velocity \(v\):\[ p = m \cdot v \]Calculate the momentum for each car:- Station wagon: \(p_1 = 1200 \times 12 = 14400\, \mathrm{kg\, m/s}\)- Second car: \(p_2 = 1800 \times 20 = 36000\, \mathrm{kg\, m/s}\)Add these to find the total momentum:\[ p_{\text{total}} = p_1 + p_2 = 14400 + 36000 = 50400\, \mathrm{kg\, m/s} \]

Step 3: Calculate the Speed of the Center of Mass

The speed of the center of mass \(v_{\text{cm}}\) can be found using the total momentum from Step 2 divided by the total mass of the system:\[ v_{\text{cm}} = \frac{p_{\text{total}}}{m_1 + m_2} = \frac{50400}{1200 + 1800} = \frac{50400}{3000} = 16.8\, \mathrm{m/s} \]

Step 4: Verify the Total Momentum Using Center of Mass Speed

Use the speed of the center of mass to verify the total momentum:\[ p_{\text{cm}} = (m_1 + m_2) \cdot v_{\text{cm}} = 3000 \times 16.8 = 50400\, \mathrm{kg\, m/s} \]This matches the total momentum found in Step 2, confirming the calculations.

Key Concepts

Center of Mass

Understanding the concept of the center of mass can simplify complex problems like those involving systems of particles or cars in motion. The center of mass is essentially the weighted average position of all the mass in a system. In the context of automobiles, like in our exercise, finding the center of mass helps in predicting the motion of the entire system. To calculate it, we sum the moments (mass times position) of each object and divide by the total mass of the system. The formula is given by: \[ x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \] Here, each component's mass acts like its weight in the calculation, `pulling` the center of mass closer to its location. Thus, the center of mass is not necessarily at the physical center between the objects but rather biased towards the heavier or closer mass.

Momentum Calculation

Momentum represents the quantity of motion an object has and depends directly on both mass and velocity. It's given by the product of an object's mass and its velocity, \[ p = m \cdot v \]. In a multi-object system, like cars on a highway, total momentum can be calculated by adding the momentum of the individual cars. For this, calculate each car's momentum and add these values together. This conservation of momentum principle is crucial because it describes how the momentum before an event, like a collision, will be equal to the momentum after, assuming no external forces interfere.

• Station wagon: \( p_1 = 1200 \times 12 \ = 14400 \, \mathrm{kg \cdot m/s}\)
• Second car: \( p_2 = 1800 \times 20 \ = 36000 \, \mathrm{kg \cdot m/s} \)

The calculation of total momentum is vital in understanding how the system as a whole will behave in situations like merging traffic or accidents.

Automobile Physics

Automobile physics encompasses a variety of concepts, including forces, motion, energy, and momentum, all of which can be applied to vehicles on the road. In this context, the examination of two cars as a system and the calculation of their center of mass and momentum provides insights into how they will behave when their paths intersect. Understanding these principles is crucial for designing safer cars, predicting traffic flow, and enhancing performance. By calculating the center of mass of a pair of vehicles, engineers can forecast how the cars might react in different situations, such as acceleration or towing. Calculations also underplay the invisible forces of friction and air resistance acting on the car, which become particularly relevant at high speeds.

Kinematics

Kinematics, often thought of as the study of motion without regard to the forces that cause it, plays a vital role in understanding how vehicles move. It connects concepts like velocity, acceleration, and displacement. In our exercise, calculating the velocity of the center of mass gives us insight into the movement pattern of the entire car system, which is the cruising speed where both cars move in unison.The speed of the center of mass, calculated using the formula, \[ v_{\text{cm}} = \frac{p_{\text{total}}}{m_1 + m_2} \], ensures that you understand how the joined system behaves. Such calculations help automate systems for optimal fuel efficiency and safety features in modern cars. Besides, it prepares us for scenarios where vehicles interchange lanes, surpass one another, or engage in sudden stops, predicting distances covered and speeds achieved.