Two masses are connected by a string. Mass A is on a flat roof and mass B is hanging over the edge of the roof. The mass on the roof is 15 kg and the hanging mass is 5 kg. The edge of the roof is 100 m above the ground. The two masses are initially at rest with the hanging mass 80 m above ground and the roof mass 50 m from the edge. There is no friction.
(Allowed to use only work/energy no kinematics)
after three seconds of motion the masses have moved 11 m.
a) calculate the speed of the masses at this instant
b) calculate tension in the rope
c) at this instant the mass on the roof travels along a section with friction where u= .577. how far along the roof does this mass move before stopping (the masses are still connected)
An atwood machine without a pulley.
work done by gravity= 11m*Bg
KE gained= 1/2 (A+B)v^2
set those two equal, solve for v
b) Tension in rope: on the second mass, Tension*11=1/2 A v^2, solve for tension
c) Now, you have initial KE of 1/2 (A+B)v^2, you are gaining GPE energy at Bg v rate, and friction is eating energy at mu*A*g*v rate
Now the average velocity during the slowing period is v/2, for a duration T seconds.
Frictionenergy=GPEgained+initialKE
mu*A*g*(v/2)T=mg(v/2)T+ 1/2 (A+B)v^2
solve for T, the time to slow to a stop.
how far did it travel? (v/2)T
thank you!
To solve these problems, we will use the concepts of work and energy. Work is defined as the transfer of energy when a force is applied over a distance. The total mechanical energy of a system is the sum of its kinetic energy (KE) and potential energy (PE). In a system with no external forces at play (like in this case without friction), the total mechanical energy is conserved.
a) To calculate the speed of the masses at this instant, we can use the principle of conservation of mechanical energy. Since the total mechanical energy is conserved, we can equate the initial mechanical energy (before motion) to the final mechanical energy (after three seconds of motion):
Initial mechanical energy = Final mechanical energy
The initial mechanical energy is the sum of the potential energy (PE) and kinetic energy (KE) of the masses on the roof and hanging off the edge:
Initial mechanical energy = PE_roof_initial + KE_roof_initial + PE_hanging_initial + KE_hanging_initial
The final mechanical energy is the sum of the potential energy (PE) and kinetic energy (KE) of the masses after three seconds of motion:
Final mechanical energy = PE_roof_final + KE_roof_final + PE_hanging_final + KE_hanging_final
Since the masses are initially at rest, their initial kinetic energies are zero:
Initial mechanical energy = PE_roof_initial + PE_hanging_initial
Final mechanical energy = PE_roof_final + KE_roof_final + PE_hanging_final + KE_hanging_final
The potential energy of an object of mass m at a height h is given by PE = mgh, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Therefore, we can calculate the final potential energy of the hanging mass using:
PE_hanging_final = m_hanging * g * h_hanging_final
Given that the hanging mass is 5 kg and has moved 11 m downwards, we have:
PE_hanging_final = 5 kg * 9.8 m/s^2 * 11 m
Next, we can calculate the final potential energy of the mass on the roof using:
PE_roof_final = m_roof * g * h_roof_final
Given that the mass on the roof is 15 kg and has moved horizontally 11 m, we have:
PE_roof_final = 15 kg * 9.8 m/s^2 * 11 m
Finally, equating the initial and final mechanical energies, we can solve for the speed of the masses at this instant.
b) To calculate the tension in the rope, we can consider the forces acting on the hanging mass. The only force acting vertically is the weight of the hanging mass, which is given by:
Weight_hanging = m_hanging * g
According to Newton's second law, the net force acting on an object is equal to its mass times its acceleration:
Net force = m_hanging * a_hanging
Since the motion is vertically downward, the net force is the difference between the weight and the tension:
Net force = Weight_hanging - Tension
Therefore, we can find the tension in the rope by solving for it.
c) To calculate the distance that the mass on the roof moves before it stops, we need to consider the work done by the force of friction. The work done by friction can be calculated as the product of the frictional force and the distance traveled.
The work done by friction can be found using the equation:
Work_friction = force_friction * distance_roof
The force of friction can be found using the equation:
Force_friction = coefficient_friction * normal_force
The normal force is the force exerted by the roof on the mass and can be found using:
Normal_force = m_roof * g
Therefore, we can calculate the work done by friction using the given coefficient of friction and the mass on the roof.
Finally, the distance traveled by the mass on the roof before stopping is equal to the work done by friction divided by the work done by the external force (tension in the rope).