A 0.51-kg mass is attached to a horizontal spring with k = 108 N/m. The mass slides across a frictionless surface. The spring is stretched 26 cm from equilibrium, and then the mass is released from rest.
(a) Find the mechanical energy of the system.
(b) Find the speed of the mass when it has moved 9 cm.
(c) Find the maximum speed of the mass.
this case the energy of sistem is
conservation of energy:
(1/2)kx² = (1/2)mv² then you can find the energy of springs, this energy would be the energy of total sistem because no exist friccion.
the maximum velocity of the mass v= square root((kx²)/ m)
the speed when it has moved 9 cm
x=(26 -9) =17cm this x replaces in this formula.
enjoy the answer
To find the mechanical energy of the system, we need to consider both the potential energy and the kinetic energy of the mass-spring system.
(a) The potential energy of the system can be calculated using the equation for the potential energy of a spring:
Potential energy (PE) = (1/2)kx^2
where k is the spring constant and x is the displacement from equilibrium. In this case, k = 108 N/m and x = 26 cm = 0.26 m.
PE = (1/2)(108 N/m)(0.26 m)^2 = 3.3648 J
Since the mass is released from rest, it initially has no kinetic energy. Therefore, the total mechanical energy (ME) of the system is equal to the potential energy:
ME = PE = 3.3648 J
(b) To find the speed of the mass when it has moved 9 cm, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system remains constant throughout the motion.
At the initial position, the potential energy is given by:
PE_initial = (1/2)kx_initial^2
where x_initial is the initial displacement from equilibrium. In this case, x_initial = 26 cm = 0.26 m.
PE_initial = (1/2)(108 N/m)(0.26 m)^2 = 3.3648 J
At the position where the mass has moved 9 cm, the potential energy is given by:
PE_final = (1/2)kx_final^2
where x_final is the final displacement from equilibrium. In this case, x_final = 9 cm = 0.09 m.
PE_final = (1/2)(108 N/m)(0.09 m)^2 = 0.4374 J
Since mechanical energy is conserved, the initial potential energy is equal to the sum of the final potential energy and the final kinetic energy:
PE_initial = PE_final + KE_final
3.3648 J = 0.4374 J + KE_final
Solving for KE_final:
KE_final = 3.3648 J - 0.4374 J = 2.9274 J
The kinetic energy (KE) of the mass can be calculated using the equation:
KE = (1/2)mv^2
where m is the mass and v is the speed of the mass.
2.9274 J = (1/2)(0.51 kg)v^2
Simplifying and solving for v:
v^2 = (2 * 2.9274 J) / (0.51 kg)
v^2 = 11.4284 m^2/s^2
v = sqrt(11.4284) m/s
v ≈ 3.38 m/s
Therefore, the speed of the mass when it has moved 9 cm is approximately 3.38 m/s.
(c) To find the maximum speed of the mass, we can consider that the maximum speed occurs when all the potential energy is converted into kinetic energy. At the maximum speed, the potential energy is zero.
PE_max = 0
Therefore, the total mechanical energy at maximum speed is equal to the kinetic energy:
ME_max = KE_max
Using the equation for kinetic energy:
ME_max = (1/2)mv_max^2
Substituting the values:
3.3648 J = (1/2)(0.51 kg)v_max^2
Simplifying and solving for v_max:
v_max^2 = (2 * 3.3648 J) / (0.51 kg)
v_max^2 = 13.3247 m^2/s^2
v_max = sqrt(13.3247) m/s
v_max ≈ 3.65 m/s
Therefore, the maximum speed of the mass is approximately 3.65 m/s.