Hey, my question is:
The rope of a swing is 3.30 m long. Calculate the angle from the vertical at which a 94.0 kg man must begin to swing in order to have the same KE at the bottom as a 1550 kg car moving at 1.57 m/s (3.51 mph)
Thanks !
To solve this question, we need to apply the principle of conservation of mechanical energy. The mechanical energy of an object consists of its kinetic energy (KE) and its potential energy (PE).
The equation for mechanical energy is given by:
E = KE + PE
At the top of the swing (when the man is at the highest point), all of the gravitational potential energy is converted to kinetic energy. Therefore, the mechanical energy is only kinetic energy at this point.
At the bottom of the swing, the mechanical energy is the sum of kinetic energy and potential energy.
Now, let's break down the solution step-by-step:
1. Calculate the kinetic energy of the car:
KE_car = (1/2) * mass_car * velocity_car^2
Given the mass of the car is 1550 kg and the velocity is 1.57 m/s, we can substitute these values into the equation:
KE_car = (1/2) * 1550 kg * (1.57 m/s)^2
2. Calculate the potential energy at the bottom of the swing:
PE_bottom = mass_man * g * height
The height at the bottom of the swing is half the length of the rope, i.e., 3.30 m / 2. We also need to consider the weight of the man, which is mass_man * g, where g is the gravitational acceleration (approximately 9.8 m/s^2). Thus, the equation becomes:
PE_bottom = (mass_man * g) * (3.30 m / 2)
3. Now, we need to equate the kinetic energy of the car to the mechanical energy at the bottom of the swing and solve for the angle (θ) at which the man must start swinging:
KE_bottom = KE_car + PE_bottom
Substitute the values we calculated for KE_car and PE_bottom:
KE_bottom = (1/2) * 1550 kg * (1.57 m/s)^2 + (mass_man * g) * (3.30 m / 2)
4. Rearrange the equation to solve for the angle θ:
KE_bottom - (1/2) * 1550 kg * (1.57 m/s)^2 = (mass_man * g) * (3.30 m / 2)
Solve for mass_man:
mass_man = (KE_bottom - (1/2) * 1550 kg * (1.57 m/s)^2) / (g * (3.30 m / 2))
5. Finally, calculate the angle θ using inverse trigonometric functions:
θ = tan^(-1)((3.30 m / 2) / (mass_man * g))
Plug in the calculated value of mass_man into the equation to find the angle θ.