Question

Alice applies a force of magnitude FA on an object, and Bob applies a force of magnitude FB on the same object. When they apply their forces in the same direction (the positive x-direction) the object accelerates with 6 m/s2. When they apply their forces into opposite directions (Alice in the positive x-direction and Bob in the negative direction), the object continues to accelerate into the positive x-direction, but the acceleration is only 3 m/s2. How much larger is FA compared to FB? (Assume FA and FB are the only forces acting on the object.)

Answer 1

By applying Newton's second law of motion and the given accelerations in two different scenarios, we find that the magnitude of force FA is 3 times larger than the magnitude of force FB.

Step-by-step explanation:

According to Newton's second law of motion, the net force (Fnet) acting on an object is equal to the mass (m) of the object multiplied by its acceleration (a), which can be expressed as Fnet = ma. When Alice and Bob apply their forces in the same direction, we have the equation FA + FB = ma. When they apply their forces in opposite directions, the equation becomes FA - FB = ma, as the forces are now working against each other. From the given problem, we know two different acceleration values, 6 m/s2 and 3 m/s2 for these two scenarios.

Let's define the mass of the object being accelerated as 'm'. Using the two equations with the two different accelerations values, we have:

• 
  
• FA + FB = m * 6
• 
  
• FA - FB = m * 3

We can solve these equations simultaneously to find the values of FA and FB. Adding these two equations, we get:

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• 2FA = m * (6 + 3) = m * 9
• 
  
• FA = (m * 9) / 2

Subtracting the second equation from the first, we get:

• 
  
• 2FB = m * (6 - 3) = m * 3
• 
  
• FB = (m * 3) / 2

Dividing the equations for FA by FB:

• 
  
• FA / FB = ((m * 9) / 2) / ((m * 3) / 2)
• 
  
• FA / FB = 9 / 3 = 3

Therefore, the magnitude of force FA is 3 times larger than the magnitude of force FB.

Answer 2

The magnitude of FA is 4.5 units and the magnitude of FB is 1.5 units. Therefore, FA is 3 times larger than FB.

Step-by-step explanation:

In order to find the difference between the magnitudes of FA and FB, we need to analyze the given information. When Alice and Bob apply their forces in the same direction, the object accelerates with 6 m/s². Let's denote FA as x and FB as y, and use Newton's second law: Fnet = ma.

When Alice and Bob apply their forces in the same direction, the net force can be written as FA - FB = (ma), where a is the acceleration. Substituting the given values, we have x - y = 6. (Equation 1)

When Alice and Bob apply their forces in opposite directions, the net force becomes FA + FB = (ma), but the acceleration is now 3 m/s². So we have x + y = 3. (Equation 2)

We can now solve the system of equations (Equation 1 and Equation 2) to find the values of x and y. Adding Equation 1 and Equation 2, we get 2x = 9. Solving further, x = 4.5. Substituting the value of x in Equation 1, we get 4.5 - y = 6. Solving this equation, we find y = -1.5.

Therefore, the magnitude of FA (x) is 4.5 units, and the magnitude of FB (y) is 1.5 units. Hence, FA is 3 times larger than FB.