Question

(Step 2.6) There were two questions concerning the forces acting on the carts connected by a string that you were to ponder: [a.] Which object, the string (between the carts) or cart A, exerts a force directly on cart B? Which object, the string or cart A, has a force acting directly on It from cart B? [b.] In light of your answers to the above question, do the forces measured in steps 2.1 through 2.4 (that is, the forces registered by each force sensor during a Pull-Hold-Release sequence) qualify as a third law pair? If these forces are not third-law pairs, why would they be equal in magnitude (within the uncertainty in the measurement)? The following hypothetical experiment will help you think about these questions. Imagine that instead of a string between the two carts, you have instead a magic cart? whose mass m2 can be varied at will, all the way to zero: Assume the tension in the string is T = 2.00 N, m1 = 3.50 kg and m3 = 4.50 kg. Also assume that friction can be neglected. Get a piece of paper and pencil and use the three free body diagrams drawn above the carts to find equations for all of the forces in terms of m1, m2, m3 and T. You can use Newton?s second law to relate the forces and the acceleration of each cart, which is the same for all of the carts, and you can use the third law to relate the magnitudes of the third law pairs. From these equations calculate the values of the four forces for four different values of the mass of the magic cart (including 0 kg), and fill in the table below. Signs are important! Assume that the positive direction is to the right. Magnitudes must be correct to within 1% of the key. You should put in three significant figures to ensure that you are within tolerance.

Answer 1

Because of the force T , each mass will undergo an acceleration of a in the drirection of T, which is

T= (m₁ + m₂ + m₃) a

From FBD of m₁ : T - F₂₁ = m₁ a

From FBD of m₂ : F₁₂ - F₃₂ = m₂ a

From FBD of m₃ : F₂₃ = m₃ a

Using Newton's Third law of motion, F₁₂ and F₂₁ are equal.

Similarly, F₂₃ and F₃₂ are equal.

Hence we can find their values by using the above equations.

m2 0.4 0.04 0.167 0.238 0.248 0.25 F21 F12 F32 F23 1.415 1.415 0.752 0.752 1.167 1.167 1.071 1.071 1.132 1.132 1.116 1.116 1.125 1.125 1.125 1.125