A 25.0-kg child on a 3.00-m-long swing is released from rest when the ropes of the swing make an angle of 31.0° with the vertical.
(a) Neglecting friction, find the child's speed at the lowest position.
m/s
(b) If the actual speed of the child at the lowest position is 2.50 m/s, what is the mechanical energy lost due to friction?
J
a)mgh = 1/2 mv^2
h = 3-3cos31
solve for v
b) mgh - 1/2mv^2 = energy lost
(a) Well, it seems like the child really wanted to swing into action! Let's calculate the child's speed at the lowest position. Time for some physics fun!
First, we need to find the potential energy of the child at the highest point of the swing. Using the formula for gravitational potential energy, which is mgh, where m is the mass (25.0 kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height (which we'll calculate).
To find the height, we need to consider the angle of the swing. Since the swing is released from rest when the ropes make an angle of 31.0° with the vertical, we can find the height using trigonometry. The height is given by h = L * sin(theta), where L is the length of the swing (3.00 m) and theta is the angle (31.0°).
So, h = 3.00 m * sin(31.0°).
Next, let's find the child's speed at the lowest position using the principle of conservation of energy. At the highest position, all the potential energy is converted into kinetic energy at the lowest position. So, the potential energy at the highest point is equal to the kinetic energy at the lowest point.
So, mgh = (1/2)mv^2, where v is the speed we need to find.
Plugging in the values we know, we have:
25.0 kg * 9.8 m/s^2 * h = (1/2) * 25.0 kg * v^2.
Now we have an equation to solve for v. Let's plug in the value we found for h and solve for v.
(b) To find the mechanical energy lost due to friction, we need to calculate the total mechanical energy at the highest point of the swing, and then subtract the kinetic energy at the lowest point from it.
The mechanical energy at the highest point is given by the gravitational potential energy, which we've already calculated as mgh.
The kinetic energy at the lowest point is given by (1/2)mv^2, where m is the mass (25.0 kg) and v is the speed at the lowest point (2.50 m/s).
So, the mechanical energy lost due to friction is the difference between the gravitational potential energy and the kinetic energy at the lowest point, which is mgh - (1/2)mv^2.
Now we can plug in the values we know and calculate the mechanical energy lost due to friction.
So, let's swing into action and calculate the answers!
To solve this problem, we can apply the principles of conservation of mechanical energy. The mechanical energy of the system at any point can be expressed as the sum of the kinetic energy and potential energy.
(a) First, let's find the child's speed at the lowest point by neglecting friction. The potential energy at the highest point is given by
PE_highest = m * g * h = m * g * (1 - cos(theta)),
where m is the mass of the child, g is the acceleration due to gravity, h is the height of the swing (which can be calculated as L * (1 - cos(theta)), L being the length of the rope), and theta is the angle made by the rope with the vertical.
At the lowest point, all the potential energy is converted into kinetic energy:
KE_lowest = (1/2) * m * v^2,
where v is the child's speed at the lowest point.
Since mechanical energy is conserved, we have:
PE_highest = KE_lowest
m * g * (1 - cos(theta)) = (1/2) * m * v^2
Simplifying and solving for v, we get:
v = sqrt(2 * g * (1 - cos(theta)))
Substituting the given values:
g = 9.8 m/s^2
theta = 31.0°
v = sqrt(2 * 9.8 * (1 - cos(31.0°)))
= 4.94 m/s
Therefore, the child's speed at the lowest position is 4.94 m/s.
(b) To find the mechanical energy lost due to friction, we need to find the initial mechanical energy and subtract the final mechanical energy (with the actual speed of 2.50 m/s).
The initial mechanical energy is:
E_initial = PE_highest = m * g * (1 - cos(theta))
The final mechanical energy is:
E_final = KE_lowest = (1/2) * m * v_final^2 = (1/2) * (25.0 kg) * (2.50 m/s)^2
The mechanical energy lost due to friction is therefore:
E_lost = E_initial - E_final
Substituting the given values:
E_lost = (25.0 kg * 9.8 m/s^2 * (1 - cos(31.0°))) - (1/2) * (25.0 kg) * (2.50 m/s)^2
E_lost = 305.13 J
Therefore, the mechanical energy lost due to friction is 305.13 J.
To solve this problem, we can use the principles of energy conservation. The total mechanical energy of the swinging child consists of gravitational potential energy and kinetic energy.
(a) To find the child's speed at the lowest position, we can apply the principle of conservation of mechanical energy. At the highest position, the swing has only potential energy, given by the formula mgh, where m is the mass of the child, g is the acceleration due to gravity, and h is the height of the swing.
At the lowest position when the child reaches its maximum speed, all potential energy is converted into kinetic energy, given by the formula (1/2)mv^2, where v is the velocity of the child.
Equating these two energies, we have:
mgh = (1/2)mv^2
The mass cancels out, and we are left with:
gh = (1/2)v^2
Rearranging the equation, we can solve for v:
v = √(2gh)
First, we need to find the height of the swing. From the given information, we know that the swing is 3.00 m long, and the ropes make an angle of 31.0° with the vertical. Using trigonometry, we can find the vertical height:
h = L * sin(θ)
h = 3.00 m * sin(31.0°)
Calculate sin(31.0°) using a scientific calculator or math software, and then multiply it by 3.00 m to find the vertical height, h.
Finally, use the equation v = √(2gh) to calculate the child's speed at the lowest position.
(b) To find the mechanical energy lost due to friction, we need to consider the difference in mechanical energy between the actual speed of the child (given as 2.50 m/s) and the ideal speed calculated in part (a). The mechanical energy lost due to friction can be calculated as:
Energy_lost = (1/2) m (v_actual^2 - v_ideal^2)
where m is the mass of the child, v_actual is the actual speed at the lowest position (given as 2.50 m/s), and v_ideal is the ideal speed calculated in part (a).
Substitute the given values into the equation to find the mechanical energy lost due to friction.
I hope this helps you understand how to solve the problem! Let me know if you have any further questions.