*41. Multiple-Concept Example 3 discusses an approach to problems such as this. The ends of a thin bar are maintained at different tempera- tures. The temperature of the cooler end is 11 °C, while the temperature at a point 0.13 m from the cooler end is 23 °C and the temperature of the warmer end is 48 °C. Assuming that heat flows only along the length of the bar (the sides are insulated), find the length of the bar.

The figure shows the cross section of a wall made of three layers. The thicknesses of the layers are L1, L2 =0.650 L1, and L3 = 0.350 L1. The thermal conductivities are k1, k2 = 0.900 k1, and kg = 0.600 k1. The temperatures at the left and right sides of the wall are TH = 30 °C and Tc= -20 °C, respectively. Thermal conduction is steady. (a) What is the temperature difference AT2 across layer 2 (between the left and right sides of the layer)? If k2 were, instead, equal to 1.180 kg, (b) would the rate at which energy is conducted through the wall be greater than, less than, or the same as previously, and (c) what would be the value of AT2? k1 ko k3 TH TC L1 L2 L3 (a) AT2 = i (b) (c) AT2 = i

An aluminum cup with mass 0.34 kg holds 0.36 kg of water. Both the cup and the water have a temperature of 12.°C. If a 0.13-kg piece of copper at 62.°C is added to the cup, what is the final equilibrium temperature in °C? You may assume that the cup, water, and copper are well insulated from anything else.

An aluminum cup with mass 0.26 kg holds 0.30 kg of water. Both the cup and the water have a temperature of 15.°C. If a 0.55-kg piece of copper at 80.°C is added to the cup, what is the final equilibrium temperature in °C? You may assume that the cup, water, and copper are well insulated from anything else. (What can I do to solve this? What equations can I use?)