A horizontal slingshot consists of two light, identical springs (with spring constants of 23.7 N/m) and a light cup that holds a 1.97-kg stone. Each spring has an equilibrium length of 50 cm. When the springs are in equilibrium, they line up vertically. Suppose that the cup containing the mass is pulled to x = 0.7 m to the left of the vertical and then released. Determine
a) the system’s total mechanical energy.
Tries 0/4
b) the speed of the stone at x = 0.
a) To calculate the system's total mechanical energy, we need to consider the potential energy stored in the springs and the kinetic energy of the stone.
The potential energy stored in each spring can be calculated using the formula:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position. Since the cup is pulled to x = 0.7 m to the left of the vertical, the displacement for each spring is 0.7 m.
Potential Energy of each spring = (1/2) * 23.7 N/m * (0.7 m)^2 = 4.13 J
Since there are two identical springs, the total potential energy stored in the springs is:
Total Potential Energy = 2 * Potential Energy of each spring = 2 * 4.13 J = 8.26 J
The kinetic energy of the stone when it is released can be calculated using the formula:
Kinetic Energy = (1/2) * m * v^2
where m is the mass of the stone and v is its velocity. The stone's mass is given as 1.97 kg.
The equilibrium position is when the springs line up vertically, so at x = 0 (after it was released), the stone has displaced downwards by a distance equal to the equilibrium length of the springs, which is 50 cm or 0.5 m.
Since the stone was pulled to x = 0.7 m and released, it will oscillate between x = -0.7 m to x = 0.7 m (crossing through the equilibrium position twice). Therefore, at x = 0, it has displaced upwards by a distance of 0.7 m.
Using the principle of conservation of mechanical energy, we can equate the potential energy at the max displacement (0.7 m) to the kinetic energy at x = 0.
Potential Energy at 0.7 m = 8.26 J
Kinetic Energy at x = 0 = (1/2) * 1.97 kg * v^2
8.26 J = (1/2) * 1.97 kg * v^2
Solving for v^2:
v^2 = (2 * 8.26 J) / 1.97 kg
v^2 = 8.37 m^2/s^2
Taking the square root of both sides:
v ≈ 2.89 m/s
Therefore, the speed of the stone at x = 0 is approximately 2.89 m/s.
To calculate the system's total mechanical energy, we need to consider the potential energy and the kinetic energy of the stone.
a) Potential Energy:
When the springs are in equilibrium, they line up vertically, and the cup is pulled to the left of the vertical by x = 0.7 m. This displacement stores potential energy in the springs.
The potential energy of a spring can be calculated using the formula: PE = (1/2) * k * x^2
where PE is the potential energy, k is the spring constant, and x is the displacement from equilibrium.
In this case, each spring has a spring constant of 23.7 N/m, and the cup is pulled 0.7 m to the left. So, the potential energy stored in each spring is:
PE = (1/2) * 23.7 N/m * (0.7 m)^2 = 5.46 J
Since there are two identical springs, the total potential energy stored in the system is:
Total PE = 2 * 5.46 J = 10.92 J
b) Kinetic Energy:
When the stone is released and reaches x = 0, it has no potential energy but only kinetic energy.
The total mechanical energy is conserved, meaning the sum of potential energy and kinetic energy remains constant throughout the motion.
Since the stone starts at rest at x = 0.7 m, the potential energy is maximum, and therefore the kinetic energy is zero.
The total mechanical energy is:
Total Mechanical Energy = Total PE + Total KE
Since at x = 0, the stone has no kinetic energy, the total mechanical energy is equal to the total potential energy:
Total Mechanical Energy = 10.92 J
b) Since the stone has no kinetic energy at x = 0, it means it has not yet started moving. Hence, the speed of the stone at x = 0 is zero.