1] One paint bucket weighing 40.0 N is hanging by a massless cord from another paint bucket weighing the same amount. The two are being pulled upward with an acceleration of 1.50m/s^2 by a massless cord attached to the upper bucket. Calculate the tension in each cord.

To solve this problem, we need to calculate the tension in each cord while considering the forces acting on the buckets and their acceleration.<br /><br />1. **Calculate the mass of each bucket:**<br /><br /> The weight of each bucket is given as 40.0 N. Using the formula for weight \( W = mg \), where \( g = 9.81 \, m/s^2 \) (acceleration due to gravity), we can find the mass \( m \):<br /><br /> \[<br /> m = \frac{W}{g} = \frac{40.0 \, \text{N}}{9.81 \, m/s^2} \approx 4.08 \, \text{kg}<br /> \]<br /><br />2. **Calculate the tension in the lower cord:**<br /><br /> The lower bucket is only affected by its own weight and the tension in the lower cord. The net force on the lower bucket is given by:<br /><br /> \[<br /> T_{\text{lower}} - W = ma<br /> \]<br /><br /> Solving for \( T_{\text{lower}} \):<br /><br /> \[<br /> T_{\text{lower}} = W + ma = 40.0 \, \text{N} + (4.08 \, \text{kg})(1.50 \, m/s^2)<br /> \]<br /><br /> \[<br /> T_{\text{lower}} = 40.0 \, \text{N} + 6.12 \, \text{N} = 46.12 \, \text{N}<br /> \]<br /><br />3. **Calculate the tension in the upper cord:**<br /><br /> The upper bucket supports both its own weight and the weight of the lower bucket, plus it provides the force needed to accelerate both buckets. The net force on the upper bucket is:<br /><br /> \[<br /> T_{\text{upper}} - 2W = 2ma<br /> \]<br /><br /> Solving for \( T_{\text{upper}} \):<br /><br /> \[<br /> T_{\text{upper}} = 2W + 2ma = 2(40.0 \, \text{N}) + 2(4.08 \, \text{kg})(1.50 \, m/s^2)<br /> \]<br /><br /> \[<br /> T_{\text{upper}} = 80.0 \, \text{N} + 12.24 \, \text{N} = 92.24 \, \text{N}<br /> \]<br /><br />Therefore, the tension in the lower cord is 46.12 N, and the tension in the upper cord is 92.24 N.