Two blocks are arranged at the ends of a massless string as shown in the figure. The system starts from rest. When the 3.21 kg mass has fallen through 0.4 m, its downward speed is 1.27 m/s. The acceleration of gravity is 9.8 m/s2 .

3.21 kg (box hanging from table)
4.96 kg (box on table)
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What is the frictional force between the 4.96 kg mass and the table?
Answer in units of N.

To find the frictional force between the 4.96 kg mass and the table, we need to analyze the forces acting on the 4.96 kg mass.

1. Identify the forces:
The forces acting on the 4.96 kg mass are:
- The weight force (mg), acting vertically downward, where m is the mass and g is the acceleration due to gravity.
- The normal force (N), exerted by the table on the mass, acting vertically upward.
- The frictional force (f), acting parallel to the table's surface and opposite to the motion of the mass.

2. Determine the net force:
Since the system is at rest initially, the net force acting on the 4.96 kg mass is zero. Therefore, the frictional force (f) must be equal in magnitude but opposite in direction to the force produced by the falling 3.21 kg mass.

3. Calculate the force generated by the falling mass:
To calculate the force generated by the falling 3.21 kg mass, we need to use the equation: Force = mass × acceleration.
The mass is 3.21 kg, and the acceleration can be calculated using the final velocity, initial velocity (0 m/s because the system is at rest), and the displacement (0.4 m). We can use the formula:
v^2 = u^2 + 2as,
where v is the final velocity (1.27 m/s), u is the initial velocity (0 m/s), a is the acceleration, and s is the displacement (0.4 m).
Rearranging the formula, we have: a = (v^2 - u^2)/2s.
Plugging in the values, we get: a = (1.27^2 - 0^2)/2 × 0.4 = 1.0275 m/s^2.

Now, using the force equation, we have:
Force = mass × acceleration = 3.21 kg × 1.0275 m/s^2 = 3.295 kg m/s^2 (or 3.295 N, since kg m/s^2 = N).

4. Determine the frictional force:
Since the net force is zero, the frictional force must be equal in magnitude to the force produced by the falling mass but in the opposite direction. Therefore, the frictional force is also 3.295 N.

So, the frictional force between the 4.96 kg mass and the table is 3.295 N.

To find the frictional force between the 4.96 kg mass and the table, we need to use Newton's second law of motion.

The equation for Newton's second law is:

Fnet = m * a

Where:
Fnet is the net force
m is the mass
a is the acceleration

In this case, we can break down the forces acting on the 4.96 kg mass as follows:

1. The weight force (mg): The weight force acting on the 4.96 kg mass is given by:

Weight force = m * g

where m is the mass and g is the acceleration due to gravity.

2. The tension force (T): The tension force in the string is the force applied by the hanging mass (3.21 kg) on the system. This force is transferred to the 4.96 kg mass through the string and acts in the upward direction.

3. The frictional force (Ff): The frictional force acts in the opposite direction of motion and can be calculated using:

Ff = μ * N

where μ is the coefficient of friction and N is the normal force.

Since the 4.96 kg mass is on a horizontal table, the normal force is equal in magnitude to the weight force acting downward (mg). Therefore, we can write:

Ff = μ * mg

To find the coefficient of friction, we need to use the information given. The system starts from rest, so the net force is equal to the tension force:

Fnet = T

As the 3.21 kg mass falls through 0.4 m, its downward speed is given as 1.27 m/s. This speed is equal to the speed of the 4.96 kg mass since they are connected by the string.

Now, we can find the tension force by using the equation:

T = m * a

where m is the mass of the hanging mass (3.21 kg), and a is the acceleration of both masses.

Using the equation:

a = (vf^2 - vi^2) / (2d)

where vf is the final velocity (1.27 m/s), vi is the initial velocity (0 m/s), and d is the distance covered (0.4 m).

Once you have found the tension force, substitute it back into the equation for the frictional force to find the answer in units of N.

Use the kinematic equation v^2=u^2+2ax

When you find the acceleration multiply it by the mass of the block on the table