mass of 4.00 kg rests on a rough horizontal table. The coefficient of static friction between the block and the table is 0.54. The block is attached to a hanging mass by a string that goes over a smooth pulley, as shown in the diagram. Determine the largest mass that can hang in this way without forcing the block to slide. Use g = 9.8 m/s/
To determine the largest mass that can hang without forcing the block to slide, we need to consider the forces acting on the block.
The first force we need to consider is the force of gravity. The weight of the block is given by the formula W = m * g, where m is the mass of the block and g is the acceleration due to gravity.
W = 4.00 kg * 9.8 m/s^2
W = 39.2 N
Next, we need to consider the force of static friction. The formula for static friction is given by the formula Ff = µ * N, where µ is the coefficient of static friction and N is the normal force.
In this case, the normal force is equal to the weight of the block, since the block is at rest on the horizontal table.
N = W
N = 39.2 N
Now we can calculate the maximum force of static friction:
Ff = µ * N
Ff = 0.54 * 39.2 N
Ff = 21.168 N
The force of static friction acts in the opposite direction of the force trying to make the block slide. If the hanging mass is too large, the force due to the hanging mass will be greater than the force of static friction, and the block will start to slide.
Since the force due to the hanging mass is transmitted through the string and the pulley to the block, the maximum force due to the hanging mass that can be supported without forcing the block to slide is equal to the force of static friction.
Therefore, the largest mass that can hang in this way without forcing the block to slide is equal to the force of static friction divided by the acceleration due to gravity:
m = Ff / g
m = 21.168 N / 9.8 m/s^2
m ≈ 2.16 kg
So, the largest mass that can hang in this way without forcing the block to slide is approximately 2.16 kg.
To determine the largest mass that can hang without forcing the block to slide, we need to consider the forces acting on the block.
Let's analyze the forces:
1. Weight of the block (mg): The weight of the block is given by the mass (m) of the block multiplied by the acceleration due to gravity (g). In this case, m = 4.00 kg and g = 9.8 m/s², so the weight of the block is (4.00 kg)(9.8 m/s²) = 39.2 N.
2. Tension in the string (T): The tension in the string is the force transmitted through the string connecting the block and the hanging mass. Since the block is not moving, the tension in the string must be equal to the static friction force acting on the block.
3. Static friction force (fs): The static friction force opposes the tendency of the block to slide. Its maximum value is determined by the coefficient of static friction (μs) multiplied by the normal force (N) exerted on the block by the table.
The normal force (N) is equal to the weight of the block (mg) acting vertically upward, so N = 39.2 N.
The maximum static friction force (fs) can be calculated as fs = μs * N, where μs = 0.54.
Now, we can equate the tension in the string (T) to the maximum static friction force (fs):
T = fs
Substituting the values, we have:
T = (0.54)(39.2 N)
T ≈ 21.168 N
Now, we can determine the largest mass that can hang without forcing the block to slide.
Since the tension in the string (T) is equal to the weight of the hanging mass, we have:
T = m_hang * g
Where m_hang is the mass of the hanging mass.
Rearranging the equation, we get:
m_hang = T / g
Substituting the values, we have:
m_hang = 21.168 N / 9.8 m/s²
m_hang ≈ 2.16 kg
Therefore, the largest mass that can hang without forcing the block to slide is approximately 2.16 kg.