A horizontal slingshot consists of two light, identical springs (with spring constants of 37.7 N/m) and a light cup that holds a 1.37-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.
Is there a question here?
(a) Stretch in the spring is
L₁=sqrt(x²+L₀²) = sqrt(0.7² +0.5²)=0.86 m
Elongation is
Δx= L₁-L₀= 0.86-0.5=0.36 m.
E(total) = PE=2•k•Δx²/2=2•37.7•0.36²/2=4.9 J.
(b) PE=KE =mv²/2
v=sqrt(2•PE/m) = sqrt(2•4.9/1.37) = 2.67 m/s
A horizontal slingshot consists of two light, identical springs (with spring constants of 45.7 N/m) and a light cup that holds a 1.37-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.
To find the displacement of the cup when it reaches equilibrium after being released, we can use Hooke's Law and the principle of conservation of energy.
1. Calculate the total potential energy stored in the springs when the cup is pulled to x = 0.7 m:
- The equilibrium length of each spring is 50 cm, which is equivalent to 0.5 m.
- The displacement of the cup from equilibrium is 0.7 m to the left of the vertical.
- The total potential energy stored in the springs is given by the sum of the potential energies of each spring: U = (1/2)kx², where k is the spring constant and x is the displacement from equilibrium.
- Since the springs are identical and have the same spring constant, the total potential energy is U = 2 * (1/2)kx², where k = 37.7 N/m and x = 0.7 m.
2. Calculate the displacement of the cup when it reaches equilibrium:
- The total potential energy at maximum displacement (U_max) is equal to the total potential energy at equilibrium (U_eq), as the system is conservative and energy is conserved: U_max = U_eq.
- Set the potential energy at maximum displacement equal to the potential energy at equilibrium and solve for x: 2 * (1/2)kx_max² = 2 * (1/2)kx_eq².
- Simplifying the equation, we get x_max² = x_eq².
- Taking the square root of both sides, we find x_max = x_eq. The displacement of the cup when it reaches equilibrium is equal to its initial displacement from equilibrium.
Therefore, the displacement of the cup when it reaches equilibrium is 0.7 m to the left of the vertical.