Two blocks, A and B (with mass 50 kg and 100 kg, respectively), are connected by a string, as shown in the figure below. The pulley is frictionless and of negligible mass. The coefficient of kinetic friction between block A and the incline is μk = 0.28. Determine the change in the kinetic energy of block A as it moves from C to D, a distance of 18 m up the incline if the system starts from rest.
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To determine the change in the kinetic energy of block A as it moves from point C to point D, we need to calculate the work done by the net force on block A.
1. Calculate the gravitational force acting on block A:
- The weight of block A = mass × gravity acceleration
- Weight of block A = 50 kg × 9.8 m/s^2 = 490 N
2. Determine the net force acting on block A:
- The net force is equal to the force component parallel to the incline minus the force of kinetic friction.
- Force parallel to the incline = gravitational force × sin θ (where θ is the angle of the incline)
- Force parallel to the incline = 490 N × sin θ
- Force of kinetic friction = coefficient of kinetic friction × normal force
- Normal force = gravitational force × cos θ
- Force of kinetic friction = 0.28 × (50 kg × 9.8 m/s^2 × cos θ)
- Net force = Force parallel to the incline - Force of kinetic friction
3. Calculate the work done by the net force:
- Work done = net force × displacement
- Displacement = 18 m (given in the question)
- Work done = (Net force) × (displacement)
4. Calculate the change in kinetic energy:
- The work done is equal to the change in kinetic energy.
- Change in kinetic energy = Work done
By following these steps, you will be able to determine the change in the kinetic energy of block A as it moves from point C to point D.
To determine the change in kinetic energy of block A, we need to find the work done on block A. Work is calculated by the formula:
Work = Force * Distance * cos(θ)
In this case, the force we need to consider is the force due to friction between block A and the incline. The friction force is given by:
Friction Force = coefficient of kinetic friction * Normal force
The normal force is the force exerted by the incline on block A, which is equal to the weight of block A multiplied by the cosine of the incline angle:
Normal force = mass * acceleration due to gravity * cos(θ)
In this case, the incline is at an angle of 0 degrees (horizontal), so the cosine of the incline angle is 1.
Now, we can calculate the friction force:
Friction Force = coefficient of kinetic friction * (mass * acceleration due to gravity * cos(θ))
Next, we need to calculate the component of the force due to gravity that acts parallel to the incline. This force is given by:
Force due to gravity parallel to the incline = mass * acceleration due to gravity * sin(θ)
Since the system starts from rest, the initial velocity of block A is 0, so its initial kinetic energy is 0. Therefore, the change in kinetic energy ΔKE is equal to the work done on block A:
ΔKE = Work
Now, we can substitute the formulas for work and the friction force:
ΔKE = (Friction Force) * Distance * cos(θ)
Substituting the formulas for the friction force and the normal force:
ΔKE = (coefficient of kinetic friction * (mass * acceleration due to gravity * cos(θ))) * Distance * cos(θ)
Plugging in the given values:
ΔKE = (0.28 * (50 kg * 9.8 m/s^2 * 1)) * 18 m * 1
Simplifying:
ΔKE = (0.28 * 50 kg * 9.8 m/s^2) * 18 m
Finally, calculating the value:
ΔKE = 254.04 Joules
Therefore, the change in the kinetic energy of block A as it moves from C to D is 254.04 Joules.