Betty weighs 449 N and she is sitting on
a playground swing seat that hangs 0.55 m above the ground. Tom pulls the swing back and releases it when the seat is 0.76 m above the ground.The acceleration of gravity is 9.8 m/s^2.
How fast is Betty moving when the swing
passes through its lowest position?
Answer in units of m/s
To find the speed at which Betty is moving when the swing passes through its lowest position, we can use the principle of conservation of mechanical energy.
The total mechanical energy of an object is the sum of its kinetic energy (KE) and potential energy (PE). In this case, the swing only has gravitational potential energy and no other external forces or energies acting on it. Therefore, we can equate the initial potential energy to the final kinetic energy.
The initial potential energy is given by Betty's weight multiplied by the initial height above the ground:
PE_initial = m * g * h_initial
Where:
m = Betty's mass
g = acceleration due to gravity
h_initial = initial height above the ground
The final kinetic energy is given by one-half times Betty's mass multiplied by the square of her velocity at the lowest position:
KE_final = (1/2) * m * v^2
Since the swing passes through its lowest position, all the potential energy is converted into kinetic energy, and there is no loss of energy due to friction or other factors. Therefore, we can set PE_initial equal to KE_final:
m * g * h_initial = (1/2) * m * v^2
Simplifying the equation by cancelling the mass 'm' on both sides, we get:
g * h_initial = (1/2) * v^2
Now we can solve for the velocity 'v' by rearranging the equation:
v = √(2 * g * h_initial)
Plugging in the given values:
g = 9.8 m/s^2 (acceleration due to gravity)
h_initial = 0.55 m (initial height above the ground)
Calculating the answer:
v = √(2 * 9.8 * 0.55)
= √(10.78)
≈ 3.28 m/s
Therefore, Betty is moving at approximately 3.28 m/s when the swing passes through its lowest position.
To find the speed at the lowest position of the swing, we need to use the principle of conservation of mechanical energy. At the highest point, all of Betty's potential energy will be converted into kinetic energy, neglecting any losses due to friction or air resistance.
First, let's find Betty's potential energy at the highest point using the formula: potential energy (PE) = mass * acceleration due to gravity * height.
PE = 449 N * 0.55 m * 9.8 m/s^2 = 2401.995 J
Now, at the lowest point, all of Betty's potential energy is converted into kinetic energy. So, we can equate the potential energy at the highest point (2401.995 J) to the kinetic energy at the lowest point.
Kinetic energy (KE) = 2401.995 J
Now, let's find the kinetic energy using the formula: KE = 0.5 * mass * velocity^2.
2401.995 J = 0.5 * 449 N * velocity^2
Rearranging the equation, we get:
velocity^2 = (2401.995 J * 2) / (449 N)
velocity^2 ≈ 10.675 m^2/s^2
Taking the square root of both sides, we find:
velocity ≈ √10.675 ≈ 3.27 m/s
Therefore, Betty is moving at approximately 3.27 m/s when the swing passes through its lowest position.