A=4.2 cm = 0.045 m.
v(max) =Aω=A•sqrt(k/m)=...
(b) its maximum speed
v(max) =Aω=A•sqrt(k/m)=...
1. First, let's find the potential energy stored in the spring when the cart is displaced from its equilibrium position. The potential energy stored in a spring is given by the formula:
Potential Energy = 0.5 * k * x^2
Where k is the spring constant and x is the displacement from the equilibrium position.
Plugging in the values:
Potential Energy = 0.5 * 10.20 N/m * (0.0420 m)^2
2. Next, let's use the conservation of energy principle. At maximum speed, all of the potential energy stored in the spring will be converted into kinetic energy. So, the equation becomes:
Potential Energy = Kinetic Energy
3. Kinetic energy is given by the formula:
Kinetic Energy = 0.5 * m * v^2
Where m is the mass of the cart and v is the velocity.
4. Set the potential energy equal to the kinetic energy and solve for v:
0.5 * 10.20 N/m * (0.0420 m)^2 = 0.5 * 0.223 kg * v^2
5. Simplify the equation:
(10.20 N/m * 0.0420 m^2) = (0.223 kg * v^2)
6. Solve for v:
v^2 = (10.20 N/m * 0.0420 m^2) / 0.223 kg
v^2 = 0.1944 m^2/s^2 / 0.223 kg
v^2 ≈ 0.8726 m^2/s^2
7. Take the square root to find v:
v ≈ √(0.8726 m^2/s^2)
v ≈ 0.934 m/s
Therefore, the maximum speed of the cart is approximately 0.934 m/s.