Math  /  Algebra

Question34. (II) One 3.2kg3.2-\mathrm{kg} paint bucket is hanging by a massless cord from another 3.2kg3.2-\mathrm{kg} paint bucket, also hanging by a massless cord, as shown in Fig. 4-41. (a) If the buckets are at rest, what is the tension in each cord? (b) If the two buckets are pulled upward with an acceleration of 1.45 m/s21.45 \mathrm{~m} / \mathrm{s}^{2} by the upper cord, calculate the tension in each cord.
FIGURE 4-41 Problems 34 and 35.

Studdy Solution

STEP 1

1. Each paint bucket has a mass of 3.2kg3.2 \, \text{kg}.
2. The cords are massless.
3. We need to find the tension in each cord under two scenarios: (a) when the buckets are at rest, and (b) when the buckets are accelerating upward at 1.45m/s21.45 \, \text{m/s}^2.
4. The acceleration due to gravity is 9.8m/s29.8 \, \text{m/s}^2.

STEP 2

1. Analyze the forces when the buckets are at rest.
2. Calculate the tension in each cord when at rest.
3. Analyze the forces when the buckets are accelerating.
4. Calculate the tension in each cord when accelerating.

STEP 3

Analyze the forces when the buckets are at rest.
- For the bottom bucket: The only forces acting are the tension T2T_2 in the lower cord and the gravitational force mgmg. - For the top bucket: The forces are the tension T1T_1 in the upper cord, the tension T2T_2 from the lower cord, and its gravitational force mgmg.

STEP 4

Calculate the tension in each cord when at rest.
- For the bottom bucket at rest, T2=mgT_2 = mg. $ T_2 = 3.2 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 31.36 \, \text{N} \]
- For the top bucket at rest, T1=T2+mgT_1 = T_2 + mg. $ T_1 = 31.36 \, \text{N} + 31.36 \, \text{N} = 62.72 \, \text{N} \]

STEP 5

Analyze the forces when the buckets are accelerating.
- For the bottom bucket: The forces are the tension T2T_2 in the lower cord and the net force due to its acceleration mama. - For the top bucket: The forces are the tension T1T_1 in the upper cord, the tension T2T_2 from the lower cord, and the net force due to its acceleration mama.

STEP 6

Calculate the tension in each cord when accelerating.
- For the bottom bucket: $ T_2 = m(g + a) = 3.2 \, \text{kg} \times (9.8 \, \text{m/s}^2 + 1.45 \, \text{m/s}^2) = 3.2 \, \text{kg} \times 11.25 \, \text{m/s}^2 = 36.0 \, \text{N} \]
- For the top bucket: $ T_1 = T_2 + m(g + a) = 36.0 \, \text{N} + 36.0 \, \text{N} = 72.0 \, \text{N} \]
The tension in each cord when the buckets are at rest is T1=62.72NT_1 = 62.72 \, \text{N} and T2=31.36NT_2 = 31.36 \, \text{N}. When the buckets are accelerating, the tension in each cord is T1=72.0NT_1 = 72.0 \, \text{N} and T2=36.0NT_2 = 36.0 \, \text{N}.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord