The cable of a 1800-kg elevator has broken, and the elevator is moving downward at a steady speed of 1.6 m/s. A safety braking system that works on friction prevents the downward speed from increasing.
a. at what rate does mechanical energy turn into thermal energy. I found this:
28.2528 kW
b.While the elevator is moving downward at 1.6 m/s, the braking system fails and the elevator is in free-fall for a distance of 5.3 m before hitting the top of a large safety spring with force constant of 1.50 104 N/m. After the elevator hits the top of the spring, find the distance d that the spring is compressed before the elevator is brought to rest.
see other post.
a. To find the rate at which mechanical energy turns into thermal energy, we need to calculate the power dissipated as heat. Power is given by the equation:
Power = Force x Velocity
In this case, the force that acts on the elevator due to friction is equal to the weight of the elevator, which is given by the equation:
Force = Mass x Gravity
where the mass of the elevator is 1800 kg and the acceleration due to gravity is approximately 9.8 m/s^2.
Substituting the values into the equation, we get:
Force = 1800 kg x 9.8 m/s^2 = 17,640 N
The power dissipated as heat is then calculated by:
Power = Force x Velocity = 17,640 N x 1.6 m/s = 28,224 W = 28.224 kW
After rounding to four significant figures, the rate at which mechanical energy turns into thermal energy is approximately 28.2528 kW.
b. To find the distance the spring is compressed, we need to apply the principle of conservation of mechanical energy. Before the elevator hits the spring, it has gravitational potential energy due to its height, which will be converted into elastic potential energy of the spring when compressed.
The initial gravitational potential energy (PEg) of the elevator is given by the equation:
PEg = Mass x Gravity x Height
Substituting the values, we get:
PEg = 1800 kg x 9.8 m/s^2 x 5.3 m = 91,308 J
The final elastic potential energy (PEe) of the spring when compressed can be calculated using Hooke's Law:
PEe = 0.5 x Spring Constant x Compression Distance^2
Substituting the values, we get:
PEe = 0.5 x 1.50 x 10^4 N/m x d^2
Since the elevator comes to a rest when the spring is fully compressed, the final kinetic energy (KEf) is zero. Thus, the total mechanical energy (PEg + PEe + KEf) is conserved.
Setting the initial gravitational potential energy equal to the final elastic potential energy, we have:
PEg = PEe + KEf
Substituting the values and solving for d, we get:
91,308 J = 0.5 x 1.50 x 10^4 N/m x d^2
Simplifying the equation, we find:
d^2 = 91,308 J / (0.5 x 1.50 x 10^4 N/m) = 12.0872 m^2
Taking the square root of both sides, we get:
d = √(12.0872 m^2) = 3.478 m
After rounding to three significant figures, the spring is compressed by approximately 3.478 m before the elevator is brought to rest.