The two blocks (m = 13 kg and M = 89 kg) are not attached to each other. The coefficient of static friction between the blocks is μs = 0.50, but the surface beneath the larger block is frictionless. What is the minimum magnitude of the horizontal force (F) required to keep the smaller block from slipping down the larger block?
Well, I guess this is a classic case of "don't slip, little block!" Now, let's see what we can do here.
The first thing we need to know is the maximum force of static friction between the two blocks, which can be found using the formula:
Ffriction = μs × Normal force
Since the two blocks are not attached, the normal force on the smaller block is the force of gravity acting on it, which is given by:
Normal force = m × g
So, substituting the values and plugging everything in, we get:
Ffriction = μs × (m × g)
Now, the force required to keep the smaller block from slipping down is equal to the force of static friction acting in the opposite direction. In other words:
F = - Ffriction
So, let's calculate it:
F = - (μs × (m × g))
Now we can plug in the values. Assuming the acceleration due to gravity is approximately 9.8 m/s^2, we have:
F = - (0.50 × (13 kg × 9.8 m/s^2))
Calculate that out, and you'll find the minimum magnitude of the horizontal force needed to prevent the smaller block from slipping down the larger one.
But, hey, it's just physics trying to keep those blocks in line, right? Keep it steady, F!
To determine the minimum magnitude of the horizontal force required to keep the smaller block from slipping down the larger block, we can analyze the forces acting on the system.
First, let's start by calculating the maximum static friction force (F_max) between the two blocks. The formula for static friction is:
F_friction = μs * N
Where:
- F_friction is the friction force
- μs is the coefficient of static friction
- N is the normal force
Since the surface beneath the larger block is frictionless, the normal force on the smaller block is equal to its weight, which is given by:
N = m * g
Where:
- m is the mass of the smaller block
- g is the acceleration due to gravity (which is approximately 9.8 m/s²)
Thus, the maximum static friction force (F_max) is:
F_max = μs * N
= μs * (m * g)
Now, let's calculate the weight of the smaller block:
Weight = m * g
Next, we need to consider the forces acting on the smaller block. Since it is on an inclined plane (the larger block), there are two main forces acting on it:
1. Force component parallel to the inclined plane (F_parallel): This force is directed up the incline and is the force we want to determine.
2. Weight component parallel to the inclined plane (Weight_parallel): This force is directed down the incline.
To prevent the smaller block from slipping, the force parallel to the incline (F_parallel) must be equal to or greater than the weight component (Weight_parallel). Thus, we have:
F_parallel ≥ Weight_parallel
To calculate the weight component parallel to the inclined plane (Weight_parallel), we need to find the angle of the incline (θ). This can be done using trigonometry. The angle can be found by taking the inverse sine of the ratio of the height and length of the inclined plane:
θ = arcsin(height / length)
Once we have the angle, we can calculate the weight component parallel to the inclined plane:
Weight_parallel = Weight * sin(θ)
Finally, to keep the smaller block from slipping, the minimum magnitude of the horizontal force (F) required must be equal to the weight component parallel to the inclined plane (Weight_parallel):
F = Weight_parallel
By following these steps and calculations, you can determine the minimum magnitude of the horizontal force (F) required to keep the smaller block from slipping down the larger block.
To find the minimum magnitude of the horizontal force (F) required to keep the smaller block from slipping down the larger block, we need to consider the forces acting on both blocks.
Let's analyze the forces acting on the smaller block (m = 13 kg) first. There are two forces acting on it:
1. The weight of the smaller block (m * g), which acts vertically downward.
2. The static friction force (f) provided by the larger block, acting horizontally in the opposite direction.
Now let's analyze the forces acting on the larger block (M = 89 kg):
1. The weight of the larger block (M * g), which acts vertically downward.
2. The normal force (N) exerted by the surface beneath the larger block, acting vertically upward.
Since the surface beneath the larger block is frictionless, there is no horizontal force acting on the larger block.
In order to prevent the smaller block from slipping down the larger block, the static friction force between the blocks needs to be strong enough to counteract the downward force component of the smaller block's weight.
The downward force component of the smaller block's weight is given by (m * g * sinθ), where θ is the angle between the horizontal and the incline on the larger block, which can be determined using trigonometry.
To find the minimum magnitude of F, which is the horizontal force required to balance the smaller block, we need to set up an equation that involves the static friction force (f) and the downward force component of the smaller block's weight (m * g * sinθ). This equation can be written as:
f = m * g * sinθ
However, we also need to consider that the static friction force has a maximum value given by the coefficient of static friction (μs) multiplied by the normal force (N). This maximum static friction force can be written as:
f_max = μs * N
Now, the normal force (N) can be determined by considering the vertical forces acting on the larger block. Since the larger block is not moving vertically, the sum of the vertical forces must be zero. This can be represented as:
N - M * g = 0
Solving the above equation, we find:
N = M * g
Substituting this value into the equation for the maximum static friction force, we have:
f_max = μs * (M * g)
Finally, to find the minimum magnitude of F, we set it equal to the maximum static friction force, since that is the minimum horizontal force required to counteract the downward force component of the smaller block's weight:
F = f_max = μs * (M * g)
Substituting the given values μs = 0.50, M = 89 kg, and g (acceleration due to gravity) ≈ 9.8 m/s², we can calculate the minimum magnitude of F:
F = 0.50 * (89 kg * 9.8 m/s²)
Now, you can calculate the exact value for F.