A frictionless horizontal air track has a spring at either end. The spring on the left has a spring constant of 1600 N/m; the one on the right has a spring constant of 1060 N/m. A glider with a mass of 2.9 kg is pressed against the left-hand spring, compressing it by 3.7 cm. The glider is then released, so that the spring propels it rightward. It slides along the track and into the right-hand spring. What is the maximum compression of the right-hand spring?
To find the maximum compression of the right-hand spring, we can use the principle of conservation of mechanical energy.
1. First, let's find the potential energy stored in the left-hand spring when it is compressed by 3.7 cm:
- The potential energy stored in a spring is given by the formula: PE = (1/2) k x^2, where k is the spring constant and x is the displacement.
- Plugging in the values, we have PE_left = (1/2) * 1600 N/m * (0.037 m)^2
- Simplifying, we get PE_left = 0.86 J (rounded to two decimal places).
2. According to the conservation of mechanical energy, the potential energy stored in the left-hand spring is converted into the kinetic energy of the glider as it moves to the right. Thus, at the maximum compression of the right-hand spring, all the energy will be converted into potential energy.
- The potential energy stored in the right-hand spring is given by the same formula: PE = (1/2) k x^2, where k is the spring constant and x is the displacement.
- Setting PE_right = 0.86 J, we can calculate the maximum compression of the right-hand spring:
- 0.86 J = (1/2) * 1060 N/m * x^2
- Solving for x, we get x = sqrt((2 * PE_right) / k_right)
- Plugging in the values, we have x = sqrt((2 * 0.86 J) / 1060 N/m)
- Simplifying, we get x ≈ 0.052 m or 5.2 cm (rounded to two decimal places).
Therefore, the maximum compression of the right-hand spring is approximately 5.2 cm.
To find the maximum compression of the right-hand spring, we need to use the conservation of mechanical energy. The mechanical energy of the system is conserved because no external forces are doing work on the glider-spring system.
The mechanical energy of the system is given by the sum of the potential energy and kinetic energy:
E = PE + KE
Initially, when the left-hand spring is compressed, all the energy is in the form of potential energy stored in the compressed spring. When the glider is released, this potential energy is converted into kinetic energy as the glider moves rightward.
The potential energy stored in a spring is given by the formula:
PE = (1/2)kx^2
Where k is the spring constant and x is the compression or extension of the spring.
Initially, the potential energy is stored in the left-hand spring and is given by:
PE_left = (1/2)(k_left)(x_left)^2
Substituting the given values:
PE_left = (1/2)(1600 N/m)(0.037 m)^2
Simplifying, we find:
PE_left = 2.9784 J
Since mechanical energy is conserved, this potential energy is converted into kinetic energy as the glider moves rightward. When the glider reaches the right-hand spring, it compresses that spring.
The potential energy stored in the right-hand spring when it is compressed by a certain distance is given by:
PE_right = (1/2)(k_right)(x_right)^2
We want to find x_right, the maximum compression of the right-hand spring.
Given that PE_left = PE_right (conservation of mechanical energy), we can set up the following equation:
(1/2)(1600 N/m)(0.037 m)^2 = (1/2)(1060 N/m)(x_right)^2
Simplifying the equation, we find:
0.037^2(1600 N/m) = (1060 N/m)(x_right)^2
Solving for x_right, we get:
x_right = √((0.037^2(1600 N/m))/(1060 N/m))
Plugging in the values and evaluating, we find:
x_right ≈ 0.027 m (or 2.7 cm)
Therefore, the maximum compression of the right-hand spring is approximately 0.027 m.