A 2.20-kg box is sliding across the horizontal floor of an elevator. The coefficient of kinetic friction between the box and the floor is 0.480. Determine the kinetic frictional force that acts on the box when the elevator is
(a) stationary,
(b) accelerating upward with an acceleration whose magnitude is 2.60 m/s2, and
(c) accelerating downward with an acceleration whose magnitude is 2.60 m/s2.
Well, well, well, looks like we've got some physics fun going on here! Let's break it down, step by step, and add some humor along the way.
(a) When the elevator is stationary, there is no need to be extra fancy. The kinetic frictional force can be found using the formula:
frictional force = coefficient of kinetic friction × normal force
But since the elevator is stationary, the normal force is simply equal to the weight of the box. So, just calculate the normal force using the formula:
normal force = mass × gravity
where gravity is approximately 9.8 m/s².
Then you can substitute the values into the first formula to find the kinetic frictional force. And voila, you've got yourself an answer!
(b) Ah, now we have an upwardly accelerating elevator! Hang on tight, it's going to be a bumpy ride. Things get a little more exciting with the addition of this acceleration.
To find the kinetic frictional force in this case, we first need to find the net force acting on the box. Note that the net force is equal to the product of the mass and acceleration, with the added force due to friction in the opposite direction.
net force = mass × acceleration + frictional force
Substituting the known values, we can rewrite it as:
frictional force = net force - mass × acceleration
Calculate the net force by multiplying the mass by the sum of acceleration due to gravity and the acceleration of the elevator. Then, plug everything into the formula and you're good to go!
(c) Oh boy, we're going down! Time to put on your spooky clown makeup because the elevator is accelerating downward this time.
The process is basically the same as in part (b), but with a slight twist. Since the elevator is going down, the acceleration has a different sign. Remember that acceleration due to gravity is always negative. Therefore, the net force equation changes to:
net force = mass × (acceleration due to gravity - acceleration)
Again, calculate the net force and subtract the product of the mass and acceleration due to gravity. Finally, you can find the kinetic frictional force using the formula from part (a).
So, get those calculations rolling and let's see the clown show in action!
To determine the kinetic frictional force on the box in each scenario, we can use the equation:
F_friction = μ * F_normal
Where:
F_friction is the frictional force
μ is the coefficient of kinetic friction
F_normal is the normal force (equal to the weight of the box)
Let's calculate the kinetic frictional force for each scenario:
(a) When the elevator is stationary, there is no acceleration. Therefore, the normal force is equal to the weight of the box:
F_normal = m * g
F_normal = 2.20 kg * 9.8 m/s^2
F_normal = 21.56 N
Now, we can calculate the frictional force:
F_friction = 0.480 * 21.56 N
F_friction ≈ 10.35 N
Therefore, the kinetic frictional force when the elevator is stationary is approximately 10.35 N.
(b) When the elevator is accelerating upward with an acceleration of 2.60 m/s^2, the normal force on the box will be:
F_normal = m * (g + a)
F_normal = 2.20 kg * (9.8 m/s^2 + 2.60 m/s^2)
F_normal = 25.52 N
Now, we can calculate the frictional force:
F_friction = 0.480 * 25.52 N
F_friction ≈ 12.27 N
Therefore, the kinetic frictional force when the elevator is accelerating upward is approximately 12.27 N.
(c) When the elevator is accelerating downward with an acceleration of 2.60 m/s^2, the normal force on the box will be:
F_normal = m * (g - a)
F_normal = 2.20 kg * (9.8 m/s^2 - 2.60 m/s^2)
F_normal = 16.28 N
Now, we can calculate the frictional force:
F_friction = 0.480 * 16.28 N
F_friction ≈ 7.82 N
Therefore, the kinetic frictional force when the elevator is accelerating downward is approximately 7.82 N.
To determine the kinetic frictional force acting on the box in each scenario, we need to use the formula:
\(f_{k} = \mu \cdot N\)
where \(f_{k}\) is the kinetic frictional force, \(\mu\) is the coefficient of kinetic friction, and \(N\) is the normal force.
The normal force can be determined using the equation:
\(N = mg\)
where \(m\) is the mass of the box and \(g\) is the acceleration due to gravity (approximately 9.8 m/s^2).
(a) When the elevator is stationary:
Since the elevator is stationary, there is no net acceleration. Therefore, the normal force acting on the box is equal to the weight of the box. Thus, the normal force is \(N = mg = 2.20 \, \text{kg} \times 9.8 \, \text{m/s}^2\).
Now, we can calculate the frictional force using the formula:
\(f_{k} = \mu \cdot N\).
(b) When the elevator is accelerating upward:
Here, we have an additional acceleration of 2.60 m/s^2 acting in the same direction as the normal force. Therefore, the net force acting on the box is the difference between the gravitational force and the force due to the acceleration.
To calculate the normal force, we use the equation:
\(N = mg + ma\), where \(a\) is the acceleration due to the elevator.
Substituting the values, we find \(N = 2.20 \, \text{kg} \times 9.8 \, \text{m/s}^2 + 2.20 \, \text{kg} \times 2.60 \, \text{m/s}^2\).
Now, using the formula \(f_{k} = \mu \cdot N\), we can calculate the kinetic frictional force.
(c) When the elevator is accelerating downward:
Similar to the previous case, we have an additional acceleration of 2.60 m/s^2. However, this time the acceleration is in the opposite direction to the normal force. Therefore, the net force acting on the box is the sum of the forces due to gravity and acceleration.
To calculate the normal force, we use the equation:
\(N = mg - ma\), where \(a\) is the acceleration due to the elevator.
Substituting the values, we find \(N = 2.20 \, \text{kg} \times 9.8 \, \text{m/s}^2 - 2.20 \, \text{kg} \times 2.60 \, \text{m/s}^2\).
Using the formula \(f_{k} = \mu \cdot N\), we can calculate the kinetic frictional force.
By following these steps, we can find the kinetic frictional force in each scenario. Just plug in the given values and perform the necessary calculations.