Question

A horizontal slingshot consists of two light, identical springs (with spring constants of \(30.0 \mathrm{~N} / \mathrm{m}\) ) and a light cup that holds a 1.00 -kg stone. Each spring has an equilibrium length of \(50.0 \mathrm{~cm}\). When the springs are in equilibrium, they line up vertically. Suppose that the cup containing the mass is pulled to \(x=70.0 \mathrm{~cm}\) to the left of the vertical and then released. Determine a) the system's total mechanical energy. b) the speed of the stone at \(x=0\)

Short answer

Answer: The total mechanical energy of the system is 2.40 J, and the speed of the stone when x = 0 is 2.19 m/s.

Step-by-step solution

Find the potential energy in the springs

We have two springs with spring constants k = 30.0 N/m and equilibrium lengths of 50.0 cm. When the cup containing the mass is pulled to x=70.0 cm to the left of the vertical position, the springs are stretched. The potential energy U for a spring is given by the formula: \(U = \frac{1}{2}k(x - x_0)^2\) where k is the spring constant, x is the stretched length, and \(x_0\) is the equilibrium length. Since we have two springs, we need to add up the potential energies stored in both springs: \(U_\text{total} = U_1 + U_2 = \frac{1}{2}(30.0 \mathrm{~ N/m})(0.70\mathrm{~ m} - 0.50 \mathrm{~ m})^2 + \frac{1}{2}(30.0 \mathrm{~ N/m})(0.70 \mathrm{~ m} - 0.50 \mathrm{~ m})^2\)

Calculate the total potential energy

Now, plug in the values and compute the total potential energy: \(U_\text{total} = 2\left(\frac{1}{2}(30.0 \mathrm{~ N/m})(0.20 \mathrm{~ m})^2\right) = 2(30.0 \mathrm{~ N/m})(0.04 \mathrm{~ m^2}) = 2(1.20\ \mathrm{J}) = 2.40 \mathrm{J}\) The total mechanical energy of the system is the sum of the potential and kinetic energy: \(E_\text{total} = U_\text{total} + K = 2.40 \mathrm{J} + 0 = 2.40 \mathrm{J}\) (initially, the kinetic energy is zero as the stone is not moving)

Use conservation of mechanical energy

Now, we know that the total mechanical energy is conserved. When the stone reaches x = 0, the springs will be in equilibrium, and all potential energy will be transformed into kinetic energy: \(E_\text{total} = K + U(x=0)= K\) So, the kinetic energy at x = 0 is equal to the total mechanical energy: \(K = E_\text{total} = 2.40 \mathrm{J}\)

Find the speed of the stone

Finally, we can find the speed of the stone by using the kinetic energy formula: \(K = \frac{1}{2}mv^2\) where m is the mass of the stone and v is its speed. Solve for v, substituting the given values of m and K: \(v = \sqrt{\frac{2K}{m}} = \sqrt{\frac{2(2.40 \mathrm{J})}{1.00 \mathrm{~kg}}} = \sqrt{4.80 \mathrm{~ m^2/s^2}} = 2.19 \mathrm{~ m/s}\) (approx.) Now we have the answers: a) The system's total mechanical energy is \(2.40 \mathrm{J}\). b) The speed of the stone at x = 0 is \(2.19 \mathrm{~m/s}\).

Key concepts

Conservation of Mechanical Energy

The principle of the conservation of mechanical energy states that the total mechanical energy in an isolated system remains constant, as long as the system is subject to conservative forces only. This means that any energy transformed from one type to another—say, from potential energy to kinetic energy—will result in no loss of total mechanical energy.

In the context of the slingshot exercise, the two springs are pulling the stone back to the equilibrium position. As the stone is released and begins to move, the potential energy stored in the springs is converted into kinetic energy of the moving stone. Initially, all of the system's energy is potential energy. As the stone accelerates towards the equilibrium point, potential energy decreases while kinetic energy increases, yet their sum remains constant. This transformation showcases the conservation of energy where, at equilibrium, potential energy is zero and all mechanical energy is kinetic.

Potential Energy in Springs

Potential energy in springs, also known as elastic potential energy, is the energy stored in a spring when it is either compressed or stretched. This energy is directly related to the spring's displacement from its equilibrium position.

The formula to calculate the potential energy (U) in a spring is given by Hooke's Law as: \[U = \frac{1}{2}k(x - x_0)^2\]where \(k\) is the spring constant, \(x\) is the current length of the spring, and \(x_0\) is the natural length of the spring. In our exercise, the stone was pulled back, stretching the springs and thereby storing energy in them ready to be released once the stone is let go. The potential energy at any point of stretch can tell us how much energy is available for conversion to kinetic energy when the spring returns to its natural length.

Kinetic Energy Formula

Kinetic energy is the energy of motion. Any object that is moving with a certain velocity has kinetic energy. The formula to calculate the kinetic energy (K) of an object is: \[K = \frac{1}{2}mv^2\]where \(m\) is the mass of the object, and \(v\) is its velocity. In the slingshot example, when the stone is at equilibrium (\(x=0\)), all of the potential energy has been converted into kinetic energy, propelling the stone forward. By rearranging the kinetic energy formula, we can solve for velocity to find the stone's speed at the equilibrium position, illustrating how the energy has transitioned from stored to active form as the stone moves.

Spring Constant Calculation

The spring constant, denoted by \(k\), measures the stiffness of a spring and quantifies the force required to extend or compress the spring by a unit of length. It's a fundamental component in calculating both the force exerted by a spring and the potential energy stored in it.

The spring constant can be determined experimentally by applying different known forces to the spring and measuring the displacement, or it can be given as it is in this textbook problem. In our exercise, the spring constant helps us calculate the total potential energy stored in the springs when they are stretched. It's essential to remember that the spring constant is a property intrinsic to the spring, affected by the material and dimensions of the spring, and not by the force or displacement applied to it.