This experiment is testing the relationship between mass and acceleration. A string isconnecting the car (M1) to a mass (M2). Use the equation for Force F=ma to discuss;

To analyze the relationship between mass and acceleration in this setup, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (\( F = ma \)). Given that one mass (M1) is connected to another mass (M2) via a string, the forces acting on the system will need to be examined carefully.

1. What will happen to the acceleration of the car (M1) if the mass of M1 is doubled?

   Let's denote:

   • \( M1 \) as the mass of the car,
   • \( M2 \) as the hanging mass,
   • \( a \) as the acceleration of the system.

   The net force acting on the system can be described as:

   \[ F_{\text{net}} = M2 \cdot g - T \quad \text{(for mass M2)} \] \[ F_{\text{net}} = T - M1 \cdot a \quad \text{(for mass M1)} \]

   Here, \( T \) is the tension in the string and \( g \) is the acceleration due to gravity. For simplification, we consider the system in equilibrium where the acceleration is uniform.

   If the mass of \( M1 \) is doubled (let's say \( M1' = 2M1 \)), then to maintain the equilibrium, the acceleration \( a \) would decrease because the larger mass \( M1' \) requires a larger tension to maintain the same acceleration. Thus, if \( M2 \) is constant, doubling \( M1 \) will result in a decrease in acceleration \( a \).

2. What will happen to the acceleration of the car (M1) if the mass of M2 is doubled?

   If \( M2 \) is doubled (let's say \( M2' = 2M2 \)), the gravitational force acting on \( M2 \) increases. The new net force on the system would be larger because the force due to \( M2' \) now is:

   \[ F = M2' \cdot g = 2M2 \cdot g \]

   In this case, if we ignore air resistance and other frictional forces, the acceleration \( a \) of the car \( M1 \) will increase because the net force on the system increases while the mass of \( M1 \) remains the same. So, doubling \( M2 \) will lead to an increase in acceleration \( a \).

3. What will happen to the acceleration of the car (M1) if the mass of M1 AND M2 are both doubled?

   Now, if both masses are doubled (\( M1' = 2M1 \) and \( M2' = 2M2 \)), we look at the forces again. The net gravitational force acting on \( M2' \) will now be:

   \[ F = M2' \cdot g = 2M2 \cdot g \]

   However, since \( M1 \) is also doubled, we have:

   \[ \text{Mass of the car} = 2M1 \]

   Since acceleration \( a \) is a function of the net force divided by the total mass, we can see:

   \[ a = \frac{F_{\text{net}}}{M_{\text{total}}} \]

   where \( M_{\text{total}} = M1' + M2' = 2M1 + 2M2 = 2(M1 + M2) \).

   Therefore, the acceleration becomes:

   \[ a' = \frac{2M2 \cdot g}{2(M1 + M2)} = \frac{M2 \cdot g}{M1 + M2} \]

   This shows that the acceleration remains unchanged \( a' = a \) since both the forces and masses have increased proportionally. Thus, doubling both \( M1 \) and \( M2 \) keeps the acceleration \( a \) the same.

In summary:

1. The acceleration of the car (M1) decreases if \( M1 \) is doubled.
2. The acceleration of the car (M1) increases if \( M2 \) is doubled.
3. The acceleration of the car (M1) remains unchanged if both \( M1 \) and \( M2 \) are doubled.