A 0.299 kg bead slides on a curved wire, start-ing from rest at point A as described below:
A: a crest (hill) 6.15 m high
b: a valley between A and C
C: a crest (hill) 1.85 m high
The acceleration of gravity is 9.8 m/s2 .
1) If the wire is frictionless, find the speed of the bead at B. Answer in units of m/s.
2)Find the speed of the bead at C. Answer in units of m/s.
To find the speed of the bead at point B, we can use the principle of conservation of mechanical energy.
1) First, let's calculate the potential energy of the bead at point A. The formula for potential energy is given by the equation PE = mgh, where m is the mass of the bead (0.299 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the point.
PE(A) = 0.299 kg * 9.8 m/s^2 * 6.15 m
= 18.222 J
2) Next, let's calculate the potential energy of the bead at point B. Since point B is a valley, its height is 0.
PE(B) = 0
3) Now, we can use the conservation of mechanical energy to equate the initial potential energy at point A to the final sum of kinetic and potential energy at point B. Since the wire is frictionless, there is no loss of energy.
PE(A) + KE(A) = PE(B) + KE(B)
We can rearrange the equation to solve for the kinetic energy at point B:
KE(B) = PE(A) - PE(B)
= 18.222 J - 0 J
= 18.222 J
4) The formula for kinetic energy is given by KE = (1/2)mv^2, where m is the mass of the object (0.299 kg) and v is the speed of the bead at point B.
(1/2)mv^2 = 18.222 J
Solving for v:
v^2 = (2 * 18.222 J) / 0.299 kg
v^2 = 121.475 m^2/s^2
Taking the square root of both sides:
v = sqrt(121.475 m^2/s^2)
v ≈ 11.03 m/s
Therefore, the speed of the bead at point B is approximately 11.03 m/s.
To find the speed of the bead at point C, we can repeat the steps above.
1) First, let's calculate the potential energy of the bead at point C.
PE(C) = mgh
PE(C) = 0.299 kg * 9.8 m/s^2 * 1.85 m
PE(C) ≈ 5.2 J
2) Using the conservation of mechanical energy:
PE(B) + KE(B) = PE(C) + KE(C)
Since point C is a crest (hill), its height is positive.
3) Rearranging the equation to solve for KE(C):
KE(C) = PE(C) - PE(B)
KE(C) = 5.2 J - 18.222 J
KE(C) ≈ -13.022 J
Note: The negative value of KE(C) implies that the bead does not have enough energy to reach point C. Therefore, it will not reach point C and will come to a stop somewhere between point B and C on the wire.
To find the speed of the bead at point B and C, we can use the principle of conservation of energy.
1) First, let's analyze the situation at point B. From point A to B, we have a crest (hill) with a height of 6.15 m. The potential energy at point A is converted into kinetic energy at point B.
Let's calculate the potential energy at point A:
Potential energy at A = mass * gravity * height
Potential energy at A = 0.299 kg * 9.8 m/s^2 * 6.15 m
Potential energy at A = 18.13443 Joules
This potential energy is then converted into kinetic energy at point B.
Kinetic energy at B = Potential energy at A
0.5 * mass * velocity^2 = Potential energy at A
0.5 * 0.299 kg * velocity^2 = 18.13443 Joules
Now, let's solve the equation for velocity:
0.5 * 0.299 kg * velocity^2 = 18.13443 Joules
Dividing both sides of the equation by 0.1495 kg gives:
velocity^2 = (18.13443 Joules) / (0.1495 kg)
velocity^2 = 121.2575 m^2/s^2
Taking the square root of both sides gives:
velocity = √(121.2575 m^2/s^2)
velocity ≈ 11.02 m/s
Therefore, the speed of the bead at point B is approximately 11.02 m/s.
2) Now, let's analyze the situation at point C. From point B to C, we have a valley with no change in height.
The kinetic energy at point B will be conserved and converted into kinetic energy at point C. Therefore, the speed of the bead at point C will be the same as the speed at point B.
The speed of the bead at point C is approximately 11.02 m/s.
consider energy.
PE(A)=mg*6.15
KE(A)=zero, given
total energy=mg*6.15
PE(B)=0 assuming the valley is zero height
KE(B)=totalenrgy-PE(B)=mg*6.15
PE(C)=mg*1.85
KE(C)=totalenergy-PI
= mg*6.15-mg*1.85
so to find speed from ke,
v=sqrt(2*mg(height))