A block of mass 3.00 kg is pushed up against a wall by a force P that makes an angle of = 30.0° angle
with the horizontal as shown below. The coefficient of static friction between the block and the wall is
0.25. (a) Determine the possible values for the magnitude of P that allow the block to remain
stationary. (b) What happens if | P| has a larger value than | P max|? What happens if | P| has a smaller
value than |P min|? . the P is P vector
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To determine the possible values for the magnitude of force P that allow the block to remain stationary, we need to analyze the forces acting on the block.
Let's denote the force of static friction between the block and the wall as fs. The maximum static friction force can be calculated using the formula:
fs(max) = μs * N
where μs is the coefficient of static friction and N is the normal force.
The normal force N is equal to the weight of the block, which can be calculated as:
N = m * g
where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s²).
(a) To find the possible values for P, we need to equate the horizontal forces acting on the block. In this case, the horizontal forces are the component of P parallel to the wall and the force of static friction fs.
P_parallel = P * cos(θ)
fs = μs * N
Since the block is stationary, P_parallel = fs. Therefore,
P * cos(θ) = μs * N
Substituting the values into the equation:
P * cos(30°) = 0.25 * (3.00 kg * 9.8 m/s²)
Solving for P:
P = (0.25 * 3.00 kg * 9.8 m/s²) / cos(30°)
P ≈ 6.48 N
Therefore, the possible values for the magnitude of force P that allow the block to remain stationary are any values less than or equal to 6.48 N.
(b) If |P| has a larger value than |Pmax|, the block will start moving because the force of static friction will not be able to oppose the applied force P. The block will slide down the wall.
If |P| has a smaller value than |Pmin|, the block will remain stationary because the force of static friction can oppose the applied force P to keep the block in equilibrium.
To determine the possible values for the magnitude of force P that allow the block to remain stationary, we need to analyze the forces acting on the block.
Let's break down the forces acting on the block:
1. Normal Force (N): This force acts perpendicular to the wall and is equal to the weight of the block, which can be calculated as N = mg, where m is the mass of the block (3.00 kg) and g is the acceleration due to gravity (9.8 m/s^2).
2. Force due to gravity (mg): This force acts vertically downward and is given by the product of the mass (m) and the acceleration due to gravity (g).
3. Force of friction (f): This force opposes the motion of the block and acts parallel to the wall. The maximum value of static friction can be calculated as fs max = μsN, where μs is the coefficient of static friction (0.25) and N is the normal force.
4. Force P: This force is applied at an angle of θ = 30.0° with respect to the horizontal.
In order for the block to remain stationary, the force P should not exceed the static friction force, i.e., P ≤ fs max.
Now, let's calculate the magnitude of P:
1. Calculate the normal force: N = mg = (3.00 kg)(9.8 m/s^2) = 29.4 N.
2. Calculate the maximum force of static friction: fs max = μsN = (0.25)(29.4 N) = 7.35 N.
So, the magnitude of P should be less than or equal to 7.35 N for the block to remain stationary.
Answer to (a): The possible values for the magnitude of force P that allow the block to remain stationary are |P| ≤ 7.35 N.
Answer to (b): If the magnitude of P (|P|) has a larger value than |P max| (7.35 N), the block will start moving away from the wall due to a net force acting on it. If the magnitude of P has a smaller value than |P min|, the block will also start moving away from the wall due to insufficient force to overcome the static friction.