Question

rank the objects according to acceleration; use 1 for the most negative and 6 for the most positive.
directions: tap each diagram in order to toggle though the possible acceleration rankings. once each object has a unique ranking, a check answer button will appear.
2 n ← 0.25 kg → 6 n; 4 n ← 0.50 kg → 6 n; 6 n ← 4.0 kg ← 2 n; 4.0 kg → 6 n → 4 n; 8 n ← 2.0 kg → 2 n; 6 n ← 0.25 kg → 2 n

Explanation:

Step1: Recall Newton's Second Law

Newton's second law states \( F_{net} = ma \), so \( a=\frac{F_{net}}{m} \). We calculate the net force (summing forces, left as negative, right as positive) and then divide by mass for each object.

Step2: Calculate for First Object (0.25 kg, 2N left, 6N right)

Net force \( F_{net1}=6 - 2 = 4\space N \) (right). Acceleration \( a_1=\frac{4}{0.25}=16\space m/s^2 \).

Step3: Calculate for Second Object (0.50 kg, 4N left, 6N right)

Net force \( F_{net2}=6 - 4 = 2\space N \) (right). Acceleration \( a_2=\frac{2}{0.50}=4\space m/s^2 \).

Step4: Calculate for Third Object (4.0 kg, 6N left, 2N left)

Net force \( F_{net3}=-6 - 2 = -8\space N \) (left). Acceleration \( a_3=\frac{-8}{4.0}=-2\space m/s^2 \).

Step5: Calculate for Fourth Object (4.0 kg, 6N right, 4N right)

Net force \( F_{net4}=6 + 4 = 10\space N \) (right). Acceleration \( a_4=\frac{10}{4.0}=2.5\space m/s^2 \). Wait, no—wait, the diagram: 4.0 kg, two right forces? Wait, original diagram: "4.0 kg —→6N" and "—→4N"? So net force is \( 6 + 4 = 10\space N \) right. So \( a_4=\frac{10}{4}=2.5\space m/s^2 \). Wait, maybe I misread. Wait, third object: "6N ←— 4.0 kg" and "2N ←", so two left forces: 6+2=8N left, so \( F_{net}=-8\space N \), \( a_3=\frac{-8}{4}=-2\space m/s^2 \).

Step6: Calculate for Fifth Object (2.0 kg, 8N left, 2N right)

Net force \( F_{net5}=2 - 8 = -6\space N \) (left). Acceleration \( a_5=\frac{-6}{2.0}=-3\space m/s^2 \).

Step7: Calculate for Sixth Object (0.25 kg, 6N left, 2N right)

Net force \( F_{net6}=2 - 6 = -4\space N \) (left). Acceleration \( a_6=\frac{-4}{0.25}=-16\space m/s^2 \).

Now, list accelerations:
- \( a_6=-16\space m/s^2 \) (most negative, rank 1)
- \( a_5=-3\space m/s^2 \) (rank 2)
- \( a_3=-2\space m/s^2 \) (rank 3)
- \( a_4=2.5\space m/s^2 \) (rank 4)
- \( a_2=4\space m/s^2 \) (rank 5)
- \( a_1=16\space m/s^2 \) (most positive, rank 6)

Wait, wait, fourth object: maybe I messed up. Wait, fourth diagram: "4.0 kg —→6N" and "—→4N"—so both right, so net force 10N right, \( a=10/4=2.5 \). Third: two left, 8N left, \( a=-2 \). Fifth: 8N left, 2N right: net -6N, \( a=-3 \). Sixth: 6N left, 2N right: net -4N, \( a=-4/0.25=-16 \). First: 4N right, \( a=16 \). Second: 2N right, \( a=4 \).

So accelerations (from most negative to most positive):
1. Sixth object (\( a=-16 \))
2. Fifth object (\( a=-3 \))
3. Third object (\( a=-2 \))
4. Fourth object (\( a=2.5 \))
5. Second object (\( a=4 \))
6. First object (\( a=16 \))

Wait, but let's recheck fifth object: mass 2.0 kg, 8N left, 2N right. Net force: 2 - 8 = -6 N. Acceleration: -6 / 2 = -3 m/s². Correct. Sixth: 0.25 kg, 6N left, 2N right: net 2 - 6 = -4 N. Acceleration: -4 / 0.25 = -16 m/s². Correct. Third: 4.0 kg, 6N left, 2N left: net -8 N. Acceleration: -8 / 4 = -2 m/s². Correct. Fourth: 4.0 kg, 6N right, 4N right: net 10 N. Acceleration: 10 / 4 = 2.5 m/s². Correct. Second: 0.50 kg, 4N left, 6N right: net 2 N. Acceleration: 2 / 0.5 = 4 m/s². Correct. First: 0.25 kg, 2N left, 6N right: net 4 N. Acceleration: 4 / 0.25 = 16 m/s². Correct.

So ranking (1 most negative, 6 most positive):
- Sixth object: 1
- Fifth object: 2
- Third object: 3
- Fourth object: 4
- Second object: 5
- First object: 6

But the problem says "tap each diagram to toggle rankings"—but since we need to rank them, the order from most negative (1) to most positive (6) is:

1. 0.25 kg, 6N left, 2N right (sixth diagram)
2. 2.0 kg, 8N left, 2N right (fifth diagram)
3. 4.0 kg, 6N left, 2N left (third diagram)
4. 4.0 kg, 6N right, 4N right (fourth diagram)
5. 0.50 kg, 4N left, 6N right (second diagram)
6. 0.25 kg, 2N left, 6N right (first diagram)

Answer:

1. 0.25 kg (6N left, 2N right)
2. 2.0 kg (8N left, 2N right)
3. 4.0 kg (6N left, 2N left)
4. 4.0 kg (6N right, 4N right)
5. 0.50 kg (4N left, 6N right)
6. 0.25 kg (2N left, 6N right)

(Note: The answer is the ranking order of the diagrams as per the calculated accelerations, with 1 being most negative and 6 most positive.)