Two blocks connected by a string are on a horizontal frictionless surface. The blocks are connected to a hanging weight by means of a string that passes over a pulley as shown in the figure below, where m1 = 1.75 kg, m2 = 2.85 kg, and m3 = 4.95 kg.

(a) Find the tension T in the string connecting the two blocks on the horizontal surface.


(b) How much time is required for the hanging weight to fall 10.0 cm if it starts from rest?

(a) We can first find the acceleration of the whole system by considering the net force acting on it due to gravity. The net force acting on the system can be determined by the force acting on the hanging block, m3, which is m3 * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).

The equations of motion for the blocks are given by Newton's second law:

m1 * a = T
m2 * a = T
m3 * g = T + m1 * a + m2 * a

We can sum the first two equations and substitute in the third equation:

m1 * a + m2 * a = m3 * g - m1 * a - m2 * a

Solve for the acceleration a:

a = m3 * g / (m1 + m2 + m3)

Now we have the acceleration, so we can use the second equation of motion to find the tension T:

T = m2 * a

(a) Tension T:

m1 = 1.75 kg
m2 = 2.85 kg
m3 = 4.95 kg
g = 9.8 m/s^2

a = (4.95 * 9.8) / (1.75 + 2.85 + 4.95)
a ≈ 3.29 m/s^2

T = 2.85 * 3.29
T ≈ 9.38 N

The tension T in the string connecting the two blocks on the horizontal surface is approximately 9.38 N.

(b) The time t required for the hanging weight to fall 10.0 cm can be obtained from the kinematic equation of motion:

y = 1/2 * a * t^2

Since the block is falling, its distance traveled, y, is negative. Convert 10.0 cm to meters:

y = -0.1 m

Solve for t:

t = sqrt(-2 * y / a)

t = sqrt(-2 * (-0.1) / 3.29)
t ≈ 0.39 s

The time required for the hanging weight to fall 10.0 cm, starting from rest, is approximately 0.39 seconds.

To solve this problem, we can use Newton's laws of motion.

(a) To find the tension T in the string connecting the two blocks on the horizontal surface, we can consider the forces acting on each block separately.

For the first block with mass m1 = 1.75 kg, the only force acting on it is the tension T in the string. According to Newton's second law, the net force on the block is equal to its mass multiplied by its acceleration:

F_net = m1 * a1,

where a1 is the acceleration of the first block. Since the surface is frictionless, the only force acting horizontally on the block is the tension T. Therefore, we have:

F_net = T - (mass of the first block) * g = m1 * a1,

where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Now, let's consider the second block with mass m2 = 2.85 kg. The only forces acting on it are the tension T in the string, and the force due to the weight of the block, which is m2 * g. According to Newton's second law, the net force on the block is equal to its mass multiplied by its acceleration:

F_net = m2 * a2,

where a2 is the acceleration of the second block. Since the string connecting the two blocks is assumed to be inextensible and massless, the acceleration of the second block must be the same as the acceleration of the first block.

Therefore, we have:

F_net = (mass of the second block) * g - T = m2 * a2,
(m2 * g) - T = m2 * a1.

Now, we can solve these two equations simultaneously to find the tension T. First, we solve the first equation for a1:

T - (mass of the first block) * g = m1 * a1,
T = m1 * a1 + (mass of the first block) * g.

And then, we substitute this expression for T into the second equation:

(m2 * g) - (m1 * a1 + (mass of the first block) * g) = m2 * a1.

Simplifying the equation, we get:

(m2 - m1) * a1 = (m2 * g) - (mass of the first block) * g,
a1 = ((m2 * g) - (mass of the first block) * g) / (m2 - m1).

Finally, we can substitute this value of a1 back into the equation for T:

T = m1 * a1 + (mass of the first block) * g.

Substituting the given values of m1, m2, and g into the equation, we can calculate the tension T.

(b) To find the time required for the hanging weight to fall 10.0 cm, we can consider the motion of the hanging weight as it falls under the influence of gravity.

The formula for the distance fallen by an object under constant acceleration is given by:

d = (1/2) * a * t^2,

where d is the distance fallen, a is the acceleration, and t is the time.

In this case, the object starts from rest, so its initial velocity is zero. Therefore, we can simplify the formula to:

d = (1/2) * a * t^2.

Since we know the distance fallen (10.0 cm = 0.10 m) and the acceleration due to gravity (9.8 m/s^2), we can solve this equation for t.

To solve this problem, we can use Newton's second law and the equations of motion.

(a) To find the tension T in the string connecting the two blocks on the horizontal surface, we need to consider the forces acting on each block.

For m1:
- The force of tension T is directed to the right.
- The force of tension T is equal in magnitude to the force of tension acting on m2 in the opposite direction.
- The force of gravity acting on m1 is m1 * g, where g is the acceleration due to gravity.
- The net force on m1 is the difference between the force of tension and the force of gravity, which is T - m1 * g.

For m2:
- The force of tension T is directed to the left.
- The force of gravity acting on m2 is m2 * g.
- The net force on m2 is the difference between the force of tension and the force of gravity, which is T - m2 * g.

Since the two blocks are connected by a string, their accelerations must be the same. Therefore, the net force on each block must be equal.

Setting the net forces on m1 and m2 equal to each other, we have:

T - m1 * g = T - m2 * g

Simplifying the equation, we get:

m1 * g = m2 * g

Canceling out the g term, we have:

m1 = m2

Substituting the given mass values:

1.75 kg = 2.85 kg

Since this equation is not true, it means that the assumption of equal tension in the string is incorrect. Therefore, the tension T in the string connecting the two blocks on the horizontal surface cannot be determined with the given information.

(b) To find the time required for the hanging weight to fall 10.0 cm, we can use the equations of motion.

The equation relating distance, initial velocity, time, and acceleration for an object in free fall is:

d = (1/2) * g * t^2

where d is the distance, g is the acceleration due to gravity, and t is the time.

Rearranging the equation, we have:

t = sqrt(2 * d / g)

Substituting the given values:

d = 10.0 cm = 0.10 m

g = 9.8 m/s^2

t = sqrt(2 * 0.10 m / 9.8 m/s^2)

t = sqrt(0.0204)

t ≈ 0.143 s

Therefore, it takes approximately 0.143 seconds for the hanging weight to fall 10.0 cm if it starts from rest.