a block B (m2 = 14 kg) is at rest on top of a table. Block B is connected to block A (m1 = 4.8 kg) and block C (m3 = 17 kg) by two strings that each pass over a frictionless pulley installed at the either end of the table. The horizontal surface of the table exerts a 32-N force of friction on m2. If the system is released from rest, use energy concepts to find the speed of m3 after it moves down 3.0 m.
To find the speed of block C (m3) after it moves down 3.0 m, we can use the concept of conservation of energy.
Step 1: Identify the types of energies involved in the system
In this system, there are two types of energies involved - potential energy and kinetic energy.
Step 2: Calculate the initial potential energy of block C
The initial potential energy of block C (m3) can be calculated using its mass and the height it starts from. However, since block C is at rest on the table initially, the initial potential energy is zero.
Step 3: Calculate the change in potential energy of block C
The change in potential energy of block C as it moves down 3.0 m can be calculated using the formula ΔPE = mgh, where ΔPE represents the change in potential energy, m represents the mass of block C, g represents the acceleration due to gravity, and h represents the vertical distance or height the block moves down.
Given:
m3 = 17 kg
h = 3.0 m
ΔPE = (17 kg) * (9.8 m/s^2) * (3.0 m)
ΔPE = 499.8 J
Step 4: Calculate the work done against friction
The work done against friction can be calculated using the frictional force and the distance block C moves. Since the frictional force is given as 32N and block C moves 3.0 m, the work done against friction is given by the formula W = F * d, where F represents the force, and d represents the distance.
Given:
F = 32 N
d = 3.0 m
W = (32 N) * (3.0 m)
W = 96 J
Step 5: Calculate the final kinetic energy of block C
The final kinetic energy of block C can be calculated by subtracting the work done against friction (W) and the change in potential energy (ΔPE) from the initial potential energy.
Initial potential energy = 0 J
Final potential energy = Initial potential energy - ΔPE - W
Step 6: Use the conservation of energy to find the velocity (speed)
According to the law of conservation of energy, the total mechanical energy in the system is conserved. Therefore, the sum of initial potential energy and initial kinetic energy is equal to the sum of final potential energy and final kinetic energy.
Initial potential energy + Initial kinetic energy = Final potential energy + Final kinetic energy
Since the system starts from rest, the initial kinetic energy is zero.
0 J + 0 J = 0 J + Final kinetic energy
So, Final kinetic energy = 0 J
Since kinetic energy (KE) is given by the formula KE = 0.5 * m * v^2, where m represents the mass of the object, and v represents the velocity (speed), we can write
0.5 * (17 kg) * v^2 = 0 J
Simplifying, we get
8.5 * (v^2) = 0
Since v^2 cannot be negative, the only solution is v = 0 m/s.
Therefore, the speed of block C after it moves down 3.0 m is 0 m/s, it comes to rest again.
To find the speed of m3 after it moves down 3.0 m, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system is conserved, assuming no external forces act on it.
1. First, let's find the potential energy of m3 before it starts moving. The potential energy (PE) is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. In this case, h is equal to 3.0 m, and m3 has a mass of 17 kg. Therefore, the initial potential energy of m3 is PE = 17 kg * 9.8 m/s^2 * 3.0 m.
2. Next, let's find the work done by the frictional force on m2. The work done (W) by the frictional force is given by the equation W = F * d, where F is the force and d is the distance. In this case, F is equal to 32 N, and d is the distance covered by m2. Since m2 is at rest, the distance covered by m2 is equal to zero. Therefore, the work done by the frictional force is W = 32 N * 0 m.
3. The work done by the tension in the string connecting m3 to the pulley is equal to the change in kinetic energy (KE) of m3. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. Therefore, we can write W = ΔKE.
4. Since the system is released from rest, the initial kinetic energy of m3 is zero. Therefore, the work done by the tension in the string is equal to the final kinetic energy of m3. We can now equate these two values: ΔKE = 17 kg * v^2 / 2, where v is the final velocity of m3.
5. Finally, we can solve for v. Rearrange the equation to solve for v: v = sqrt(2 * ΔKE / m3), where ΔKE is equal to the initial potential energy of m3 minus the work done by the frictional force.
Substituting the given values and solving the equation, we can find the value of v.