Two blocks are connected by a lightweight, flexible cord that passes over a frictionless pulley. If m1 is 3.4 kg and m2 is 9.1 kg, and block 2 is initially at rest 130 cm above the floor, how long does it take block 2 to reach the floor?
To solve this problem, we can use the principles of Newton's laws of motion and apply the concept of conservation of energy.
First, let's determine the gravitational potential energy of block 2 when it is 130 cm above the floor. The formula for gravitational potential energy is given by:
PE = mgh
Where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
For block 2, m2 = 9.1 kg and h = 130 cm = 1.3 m. Plugging in these values into the equation, we get:
PE2 = (9.1 kg) * (9.8 m/s^2) * (1.3 m) = 113.374 J (Joules)
The potential energy gained by block 2 as it falls is used to accelerate both blocks. Since the cord is lightweight and the pulley is frictionless, there is no loss of energy during the transfer.
Now, let's find the change in potential energy of block 2 when it reaches the floor. The change in potential energy is given by:
ΔPE = PEf - PEi
Where ΔPE is the change in potential energy, PEf is the final potential energy, and PEi is the initial potential energy. In this case, the floor is considered to be the reference point, so the final potential energy is zero.
ΔPE = 0 - 113.374 J = -113.374 J
Since the potential energy decreases, the change is negative.
Now, we can equate the change in potential energy with the kinetic energy gained by block 2. The formula for kinetic energy is given by:
KE = (1/2)mv^2
Where KE is the kinetic energy, m is the mass of the object, and v is the velocity.
For block 2, we assume it starts from rest and reaches the floor with a velocity v. So the kinetic energy gained by block 2 is:
KE2 = (1/2) * (9.1 kg) * v^2 = (4.55 kg) * v^2
Since there is no friction, the kinetic energy gained by block 2 is equal to the change in potential energy:
KE2 = ΔPE
Therefore:
(4.55 kg) * v^2 = -113.374 J
We already know that v^2 is negative because it is multiplied by a negative value. Therefore, we can remove the negative sign and solve for v:
v^2 = (-113.374 J) / (4.55 kg)
v^2 ≈ -24.899 m^2/s^2
Taking the square root of both sides to find v:
v ≈ -4.989 m/s
The negative sign indicates that the velocity of block 2 is downward.
Finally, we can use the formula for constant acceleration to find the time it takes for block 2 to reach the floor:
v = at
Where v is the velocity, a is the acceleration, and t is the time taken.
In this case, the acceleration of block 2 is the acceleration due to gravity since it is experiencing free fall. So:
-4.989 m/s = (9.8 m/s^2) * t
Solving for t:
t ≈ -0.51 s
The negative sign indicates that the direction of motion is downward.
Therefore, it takes approximately 0.51 seconds for block 2 to reach the floor.
To find the time it takes for block 2 to reach the floor, we can use the concept of gravitational potential energy.
First, let's calculate the gravitational potential energy of block 2 when it's 130 cm above the floor. The formula for gravitational potential energy is:
PE = m * g * h
Where:
- PE is the potential energy
- m is the mass of the object
- g is the acceleration due to gravity
- h is the height
Given that the mass of block 2 (m2) is 9.1 kg and the height (h) is 130 cm (which is 1.3 m), we have:
PE = 9.1 kg * 9.8 m/s^2 * 1.3 m
PE = 118.604 J (Joules)
Next, since the cord is lightweight and there is a frictionless pulley, we can assume that there is no loss of energy as block 2 descends. Therefore, the potential energy is converted entirely into kinetic energy.
KE = 1/2 * m * v^2
Where:
- KE is the kinetic energy
- m is the mass of the object
- v is the velocity
Since block 2 is initially at rest, the kinetic energy is zero. Therefore, we can set the potential energy equal to zero:
PE = KE
118.604 J = 1/2 * 9.1 kg * v^2
To solve for v, we rearrange the equation:
v^2 = (2 * PE) / m
v^2 = (2 * 118.604 J) / 9.1 kg
v^2 = 26.087 J/kg
Taking the square root of both sides:
v ≈ 5.108 m/s
Now, we can use the equation of motion to find the time it takes for block 2 to reach the floor:
v = a * t
Where:
- v is the final velocity (5.108 m/s)
- a is the acceleration (which is equal to the acceleration due to gravity, 9.8 m/s^2)
- t is the time
Rearranging the equation:
t = v / a
t = 5.108 m/s / 9.8 m/s^2
t ≈ 0.521 s
Therefore, it takes approximately 0.521 seconds for block 2 to reach the floor.
Note: the two blocks are suspended vertically by the string passing over a pulley.
m1=3.4 kg
m2=9.1 kg
H=1.3 m
Let tension = T
acceleration = a
acceleration due to gravity = g
Consider block 1:
Net upward force, F1 = T-m1.g
F1.a = T-m1.g
Consider block 2:
Net downward force, F2 = m2.g - T
F2.a = m2.g - T
Solve for T and A.
Use kinematics equation to solve for time: H=v0.t + (1/2)at²
where v0 = initial velocity=0 m/s, and
t=time in seconds.