A block of ice at 0°C whose mass initially is m = 48.4 kg slides along a horizontal surface, starting at a speed vo = 4.91 m/s and finally coming to rest after traveling a distance d = 12.28 m. Compute the mass of ice melted as a result of the friction between the block and the surface. (Assume that all the heat generated owing to friction goes into the block of ice.)
Initial KE+ initial PE= final KE+finalPE+heat released
calculate heat released, initial/final PE are equal.
Now use that heat
heat= mass*heatfusion calculate mass melted.
To compute the mass of ice melted as a result of the friction, we need to calculate the work done by friction and convert it into heat energy, and then convert the heat energy into mass using the specific latent heat of fusion for ice.
1. First, let's calculate the work done by friction. The work done by friction can be given by the equation:
Work = Force x Distance
Since the force of friction is opposing the motion, it can be expressed as:
Friction Force = -Coefficient of Friction x Normal Force
Here, the normal force is equal to the weight of the ice block, which can be calculated as:
Normal Force = mass x gravity
where gravity is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Now we can substitute the values given in the problem:
mass = 48.4 kg
vo = 4.91 m/s
d = 12.28 m
Coefficient of Friction = unknown
3. Let's calculate the normal force:
Normal Force = mass x gravity
= 48.4 kg x 9.8 m/s^2
4. Next, we'll use the work-energy principle to find the work done by friction. The work done by friction can be expressed as:
Work = Change in Kinetic Energy
Since the block comes to rest, the change in kinetic energy is:
Change in Kinetic Energy = 0.5 x mass x (final velocity)^2 - 0.5 x mass x (initial velocity)^2
5. Now we can substitute the values given in the problem and solve for the coefficient of friction:
Final velocity = 0 (since the block comes to rest)
Initial velocity = 4.91 m/s
Change in Kinetic Energy = 0.5 x 48.4 kg x (0 m/s)^2 - 0.5 x 48.4 kg x (4.91 m/s)^2
Work = -Change in Kinetic Energy
Friction Force x Distance = -Change in Kinetic Energy
(-Coefficient of Friction x Normal Force) x 12.28 m = -0.5 x 48.4 kg x (4.91 m/s)^2
6. To isolate the coefficient of friction, we rearrange the equation:
Coefficient of Friction = (0.5 x mass x (initial velocity)^2) / (Normal Force x distance)
Coefficient of Friction = (0.5 x 48.4 kg x (4.91 m/s)^2) / (48.4 kg x 9.8 m/s^2 x 12.28 m)
7. Now we can calculate the coefficient of friction using the equation:
Coefficient of Friction = (0.5 x 48.4 kg x (4.91 m/s)^2) / (48.4 kg x 9.8 m/s^2 x 12.28 m)
8. Once we have the coefficient of friction, we can calculate the work done by friction:
Work = Friction Force x Distance
Work = (-Coefficient of Friction x Normal Force) x 12.28 m
9. To convert the work done by friction into heat energy, we need to multiply by -1. The negative sign indicates that the energy is being transferred into the system (the ice block).
Heat Energy = -Work
10. Finally, to convert the heat energy into mass, we divide by the specific latent heat of fusion for ice. The specific latent heat of fusion for ice is approximately 334,000 J/kg.
Mass of Ice Melted = Heat Energy / Specific Latent Heat of Fusion for Ice
Note: Make sure to use the correct units throughout the calculations to obtain the final answer in kilograms.
To solve this problem, we need to use the concepts of work, energy, and the conservation of energy.
First, let's calculate the initial kinetic energy of the block. The formula for kinetic energy is given by:
KE = (1/2) * m * v^2
where m is the mass and v is the velocity of the block.
Plugging in the values given, we have:
KE = (1/2) * 48.4 kg * (4.91 m/s)^2
= 0.5 * 48.4 kg * 24.1 m^2/s^2
= 586.372 J
Next, let's calculate the work done by friction. The work done by friction is equal to the change in kinetic energy of the block:
Work = KE - 0
Since the block comes to rest, the final kinetic energy is zero, so the work done by friction is equal to the initial kinetic energy:
Work = 586.372 J
Now, let's use the work-energy principle to find the mass of ice melted. The work done by friction equals the change in thermal energy (heat) of the block:
Work = Change in thermal energy
We can express the change in thermal energy as the mass of ice melted (melted mass) times the specific heat of ice (C) times the change in temperature (ΔT):
Work = melted mass * C * ΔT
Since the change in temperature is from 0°C to the melting point of ice (0°C), ΔT equals 0°C, so the change in thermal energy and work done by friction are both zero.
Therefore, we can set up the equation:
586.372 J = melted mass * C * 0°C
Since the specific heat of ice (C) is known to be 2.09 J/g°C, we can convert the mass from kilograms to grams (1 kg = 1000 g):
586.372 J = melted mass * 2.09 J/g°C * 0°C
586.372 J = melted mass * 2.09 J/g
Now, we can solve for the mass of ice melted:
melted mass = 586.372 J / (2.09 J/g)
melted mass = 280.34 g
Therefore, the mass of ice melted as a result of the friction between the block and the surface is 280.34 grams.