A block of weight w slides down a rough inclined plane of angle θ. A constant frictional force fk acts on the block so that it moves at a constant velocity v down the incline. Find... A block of weight w slides down a rough inclined plane of angle θ. A constant frictional force fk acts on the block so that it moves at a constant velocity v down the incline. Find the work done by each force acting on the block and the total work done on it as it descends a height h down the incline.
Understand the Problem
The question is asking for the calculation of the work done by different forces acting on a block sliding down an inclined plane, considering it moves at a constant velocity. This involves understanding the forces at play, such as gravity, friction, and any normal force, and applying work-energy principles to find the total work done as the block descends a height h.
Answer
The total work done by friction is equal to the gravitational work when the block descends a height $h$, expressed as $W = m \cdot g \cdot h$.
Answer for screen readers
The total work done by friction is equal in magnitude to the work done by gravity when the block moves at constant velocity down the incline, given by:
$$ W = m \cdot g \cdot h $$
Steps to Solve
- Identify the Forces Acting on the Block
The forces acting on the block include gravitational force, normal force, and frictional force.
- Calculate the Gravitational Force
The gravitational force acting on the block can be calculated using the formula:
$$ F_g = m \cdot g $$
where:
- $ m $ is the mass of the block
- $ g $ is the acceleration due to gravity (approximately $ 9.81 , \text{m/s}^2 $)
- Determine the Normal Force
The normal force can be found using the angle of the incline, $\theta$. It can be calculated as:
$$ F_N = m \cdot g \cdot \cos(\theta) $$
- Calculate the Frictional Force
The frictional force opposing the motion can be calculated using the coefficient of friction, $\mu$, as follows:
$$ F_f = \mu \cdot F_N $$
- Apply the Work-Energy Principle
Since the block moves at a constant velocity, the net work done is zero. Thus, the work done by gravity is equal to the work done against friction:
$$ W_g = W_f $$
The work done by gravitational force as the block descends a height $ h $ is calculated as:
$$ W_g = F_g \cdot h $$
And the work done against friction is:
$$ W_f = F_f \cdot d $$
where $ d $ is the distance traveled along the incline.
- Set the Work Done Equal
Since the net work is zero, we can express this as:
$$ F_g \cdot h = F_f \cdot d $$
From here, we can solve for the requested variable as needed.
The total work done by friction is equal in magnitude to the work done by gravity when the block moves at constant velocity down the incline, given by:
$$ W = m \cdot g \cdot h $$
More Information
In this problem, when a block slides down an incline at constant velocity, the forces acting are balanced. The work done by the gravitational force is completely countered by the work done against friction, illustrating the concept of energy conservation in mechanics.
Tips
- Forgetting to account for all forces acting on the block, especially the frictional force.
- Confusing the direction of the gravitational force and the frictional force.
- Neglecting to use the correct values for mass and height when computing work.
