a box slides under gravity down a flat slope 3 m long. Coefficient of sliding friction is 0.2.
Show that the speed of the box at the bottom of the slope is given by sqrt(6g(sin - 0.2 cos)
Thanks!
To find the speed of the box at the bottom of the slope, we can use the principles of energy conservation. Let's break down the problem step by step.
Step 1: Calculate the height difference
The box slides down a flat slope that is 3 m long. The vertical height difference (h) can be calculated using the Pythagorean theorem. Since the slope is flat, the height difference is equal to the vertical distance between the starting and ending points. So, h = 0.
Step 2: Determine the work done by gravity
The work done by gravity is equal to the change in potential energy of the box. In this case, since the height difference is zero, the work done by gravity is also zero. This is because the potential energy is directly proportional to the height difference.
Step 3: Account for friction
The sliding friction opposes the motion of the box. The work done by friction can be calculated using the equation: Work(friction) = -μk * m * g * h.
Here, μk is the coefficient of sliding friction (0.2), m is the mass of the box, and g is the acceleration due to gravity.
Since the work done by friction is negative, we can rewrite it as:
Work(friction) = μk * m * g * h
Since h = 0, the equation simplifies to:
Work(friction) = 0
Step 4: Apply conservation of energy
According to the principle of conservation of energy, the total mechanical energy of the box at any point along the slope remains constant.
Initially, the box only possesses gravitational potential energy, which is given by: PE = m * g * h. At the bottom of the slope, the box has no vertical height, so its gravitational potential energy is zero.
The kinetic energy of the box at the bottom of the slope is given by: KE = 1/2 * m * v^2, where v is the speed of the box.
Since the total mechanical energy is conserved, we can equate the initial gravitational potential energy to the final kinetic energy:
PE = KE
m * g * h = 1/2 * m * v^2
Since h = 0, the equation simplifies to:
0 = 1/2 * m * v^2
2 * 0 = m * v^2
Step 5: Solve for the speed of the box
Dividing both sides of the equation by m yields:
0 = v^2
Taking the square root of both sides:
v = 0
Hence, the speed of the box at the bottom of the slope is zero. It does not move.
If you found a different expression for the speed of the box, please check the given equation again.