The system shown in the figure below consists of a m1 = 5.52-kg block resting on a frictionless horizontal ledge. This block is attached to a string that passes over a pulley, and the other end of the string is attached to a hanging m2 = 2.76-kg block.The pulley is a uniform disk of radius 7.86 cm and mass 0.592 kg. Calculate the speed of the m2 = 2.76-kg block after it is released from rest and falls a distance of 2.08 m. Calculate the angular speed of the pulley at this instant.
m₁a= T₁
m₂a= m₂g-T₂
Iε=M => mR²•a/2R = (T₂-T₁)R,
T₂-T₁= ma/2,
a= m₂g/( m₁+m₂+m/2)=...
a=v²/2s
v=sqrt(2as)=...
ω=v/R=...
What is "s"? and what mass is the unlabeled m referring to in the first acceleration equation?
s=2.08 m
m=0.592 kg
R=0.0786 m
a is the acceleration of the system (m1+m2)
To calculate the speed of the m2=2.76-kg block after it falls a distance of 2.08 m, we can use the conservation of energy principle. The potential energy of the falling block is converted into kinetic energy.
1. Start by calculating the potential energy of the block when it is raised by 2.08 m:
Potential energy (mgh) = 2.76 kg * 9.81 m/s^2 * 2.08 m
2. Convert the potential energy to kinetic energy at the bottom:
Kinetic energy = Potential energy
3. Use the formula for kinetic energy to find the velocity of the block at the bottom:
Kinetic energy (1/2 mv^2) = Potential energy
1/2 * 2.76 kg * v^2 = mgh
Solve for v.
To calculate the angular speed of the pulley, we can use the fact that the distance the block falls is equal to the distance the string unwinds from the pulley. The linear speed of the block at the bottom is equal to the tangential speed of the outer edge of the pulley.
1. Calculate the distance the string unwinds from the pulley:
Distance = 2.08 m
2. Calculate the circumference of the pulley:
Circumference = 2 * π * radius
Use the given radius of the pulley (7.86 cm) and convert it to meters.
3. Calculate the angular distance traveled by the pulley:
Angular distance = Distance / Circumference
4. Use the formula for angular speed to find the angular speed of the pulley:
Angular speed = Linear speed / Radius
Linear speed = Distance / Time (where Time = time taken for the block to fall)
Solve for angular speed.
Now that we have the linear speed of the block and the angular speed of the pulley, we can calculate the speed of the m2=2.76-kg block after it falls and the angular speed of the pulley at that moment.