A block is at rest on a smooth horizontal surface. A constanthorizontal force begins to act on it and does work W such that theblock picks up a velocity of 128 m/s. Assume that this work W isentirely converted to heat and given to an amount of water equal tothe mass of the block. Find the subsequent rise in the temperatureof water. Specific heat capacity of water = 4096 J/Kg-C

To determine the rise in temperature of the water after the block has done work and converted that energy into heat, we can follow a systematic approach using the principles of physics. Let's break it down step by step.

Understanding Work and Energy Transfer

When a constant horizontal force acts on the block, it does work on the block, which results in the block gaining kinetic energy. The work done (W) on the block can be expressed using the formula:

• W = F × d

Where F is the force applied and d is the distance over which the force is applied. However, in this scenario, we are more interested in the kinetic energy gained by the block, which can be calculated using:

• K.E. = (1/2)mv²

Here, m is the mass of the block, and v is the final velocity of the block. Given that the block reaches a velocity of 128 m/s, we can express the work done as:

• W = (1/2)mv²

Relating Work to Heat Transfer

According to the problem, the work done on the block is entirely converted into heat, which is then transferred to an equal mass of water. The heat energy gained by the water can be calculated using the formula:

• Q = mcΔT

Where:

• Q is the heat energy absorbed by the water,
• m is the mass of the water (equal to the mass of the block),
• c is the specific heat capacity of water (given as 4096 J/kg°C),
• ΔT is the change in temperature of the water.

Calculating the Rise in Temperature

Since the work done on the block is equal to the heat gained by the water, we can set the two equations equal to each other:

• (1/2)mv² = mcΔT

We can simplify this equation by canceling the mass m from both sides (assuming the mass is not zero):

• (1/2)v² = cΔT

Now, we can solve for the change in temperature ΔT:

• ΔT = (1/2)v² / c

Substituting the values we have:

• v = 128 m/s
• c = 4096 J/kg°C

Calculating ΔT:

• ΔT = (1/2)(128 m/s)² / 4096 J/kg°C
• ΔT = (1/2)(16384 m²/s²) / 4096 J/kg°C
• ΔT = 8192 / 4096
• ΔT = 2°C

Final Result

The subsequent rise in the temperature of the water, after the block has done work and converted that energy into heat, is 2°C. This demonstrates how energy transfer works in a system where kinetic energy is converted into thermal energy, affecting the temperature of another substance.