A 25.0-kg child on a 2.00-m-long swing is released from rest when the ropes of the swing make an angle of 30.0° with the vertical. (a) Neglecting friction, find the child’s speed at the lowest position. (b) If the actual speed of the child at the lowest position is 2.00 m/s, what is the mechanical energy lost due to friction?
sketch a diagram. How how off the ground is the child at the 30 deg point:
Ans: h=2(1-cos30)
a. child speed at bottom: KE=initialPE
1/2 m v^2=mgh
v= sqrt (2gh)
b. energy lost: initial PE-1/2 m(2)^2
Well, this swing situation seems quite the ride! Let's crack some jokes as we solve it, shall we?
(a) To find the child's speed at the lowest position, we'll have to treat this problem with utmost gravity. We can break it down into two parts: the gravitational potential energy lost and the kinetic energy gained.
First, let's calculate the gravitational potential energy lost. The change in height of the child is given by the difference in the rope length, so Δh = 2.00 m. Taking the mass of the child as m = 25.0 kg and considering that g is the acceleration due to gravity at approximately 9.8 m/s², we can use the formula for gravitational potential energy: ΔPE = mgh.
Substituting the values in, we get ΔPE = 25.0 kg * 9.8 m/s² * 2.00 m * sin(30.0°). Remember, we're using the sine of the angle because it's opposite the height.
Now, let's convert this potential energy into kinetic energy. Since energy is conserved, the gravitational potential energy lost will be equal to the kinetic energy gained: ΔPE = ΔKE. Hence, ΔKE = 25.0 kg * v² / 2, where v is the velocity we want to find.
Solving for v, we get...
Wait, hold on! Why did the swing tip over?
Because it went over its wanted…
Okay, I'll stop swinging jokes around. Let's get back to the calculation.
Now, solving for v, we get v = √(2 * ΔPE / m). Plugging in the values, let's crunch the numbers and find the child's speed!
(b) Now moving on to part (b), we're told that the actual speed of the child at the lowest position is 2.00 m/s. So, we need to find the mechanical energy lost due to friction. To do this, we'll calculate the initial mechanical energy and subtract the final mechanical energy from it.
The initial mechanical energy is given by the sum of the gravitational potential energy and the kinetic energy: E_i = ΔPE + m * v² / 2.
The final mechanical energy is simply the kinetic energy: E_f = m * final velocity² / 2.
The mechanical energy lost will be the difference between the initial and final mechanical energy: E_lost = E_i - E_f.
Now, let's put on our serious faces and calculate!
And there we have it, the answers to both (a) and (b). Remember, even in the world of physics, it's always good to have a little humor swinging around!
To solve this problem, we can use energy conservation principles. The initial potential energy, when the child is released from rest, is converted into kinetic energy at the lowest position. We can then use this information to find the child's speed at the lowest position. Let's solve this step-by-step:
(a)
1. Determine the initial potential energy of the child:
The initial potential energy, U_initial, can be calculated using the formula:
U_initial = m * g * h
where m is the mass of the child (25.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the swing (2.00 m).
Plugging in the values:
U_initial = 25.0 kg * 9.8 m/s^2 * 2.00 m
U_initial = 490 J
2. Determine the kinetic energy at the lowest position:
The kinetic energy, K_final, can be calculated using the formula:
K_final = (1/2) * m * v^2
where m is the mass of the child (25.0 kg) and v is the final velocity that we need to find.
Given that there is no external work or energy losses, the initial potential energy is equal to the final kinetic energy. Therefore:
U_initial = K_final
Therefore:
(1/2) * m * v^2 = m * g * h
3. Solve for the child's final velocity:
Canceling out m from both sides and rearranging the equation, we get:
v^2 = 2 * g * h
Plugging in the given values:
v^2 = 2 * 9.8 m/s^2 * 2.00 m
v^2 = 39.2 m^2/s^2
v = √(39.2) m/s
Therefore, the child's speed at the lowest position is approximately 6.26 m/s.
(b)
Now, to find the mechanical energy lost due to friction, we can calculate the difference between the initial and final kinetic energy.
1. Determine the initial kinetic energy:
The initial kinetic energy, K_initial, can be calculated using the formula:
K_initial = (1/2) * m * (0)^2
Since the child is released from rest, the initial velocity is 0 m/s.
Therefore, K_initial = 0 J.
2. Determine the mechanical energy lost due to friction:
The mechanical energy lost due to friction, E_lost, can be calculated as the difference between the initial kinetic energy and the kinetic energy at the lowest position:
E_lost = K_initial - K_final
Therefore, E_lost = 0 J - (1/2) * m * (2.00 m/s)^2
Plugging in the given values:
E_lost = - (1/2) * 25.0 kg * (2.00 m/s)^2
E_lost = - 50 J
Since the mechanical energy cannot be negative, we take the absolute value:
E_lost = 50 J.
Therefore, the mechanical energy lost due to friction is 50 J.
To find the child's speed at the lowest position and the mechanical energy lost due to friction, we can first analyze the energy changes involved in the system.
(a) To find the child's speed at the lowest position, we can use the conservation of mechanical energy. At the highest point, the child has only potential energy, which is converted into kinetic energy at the lowest point if we neglect friction.
Step 1: Find the potential energy at the highest point:
The potential energy of the child is given by the equation: PE = mgh, where m is the mass of the child (25.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical height.
The height h can be calculated from the length of the swing and the angle made by the ropes with the vertical. We can use basic trigonometry to find the vertical displacement:
h = L * sin(theta), where L is the length of the swing (2.00 m) and theta is the angle made by the ropes with the vertical (30.0°).
Plugging in the values, we have:
h = 2.00 m * sin(30.0°)
Step 2: Calculate the potential energy:
PE = mgh = 25.0 kg * 9.8 m/s^2 * (2.00 m * sin(30.0°))
(b) To find the mechanical energy lost due to friction, we can calculate the difference between the initial mechanical energy (potential energy at the highest point) and the final mechanical energy (kinetic energy at the lowest point).
Step 3: Find the kinetic energy at the lowest position:
The kinetic energy of an object is given by the equation: KE = (1/2)mv^2, where m is the mass of the child (25.0 kg) and v is the velocity/speed at the lowest point.
Given that the speed is 2.00 m/s, we have:
KE = (1/2) * 25.0 kg * (2.00 m/s)^2
Step 4: Calculate the mechanical energy lost:
Energy lost = Initial energy - Final energy
Energy lost = PE - KE
Now that we have calculated the potential energy and kinetic energy, we can find the mechanical energy lost due to friction by subtracting the final energy from the initial energy.
Hope this helps!