As shown in the figure below, a green bead of mass 50 g slides along a straight wire. The length of the wire from point to point is 0.800 m, and point is 0.500 m higher than point . A constant friction force of magnitude 0.020 0 N acts on the bead.
(a) If the bead is released from rest at point , what is its speed at point ?
To find the speed of the bead at point B, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system is conserved, which means that the initial potential energy at point A is converted to kinetic energy at point B.
We can calculate the potential energy at point A using the formula:
Potential energy at A = mass * gravity * height
where mass = 50 g = 0.050 kg (since 1 kg = 1000 g), gravity = 9.8 m/s^2, and height = 0.500 m.
Potential energy at A = 0.050 kg * 9.8 m/s^2 * 0.500 m
= 0.245 J
The total mechanical energy at point A is the sum of the potential energy and kinetic energy:
Total mechanical energy at A = Potential energy at A + Kinetic energy at A
Since the bead is released from rest, the initial kinetic energy at point A is zero.
Total mechanical energy at A = 0.245 J + 0 J
= 0.245 J
At point B, all of the initial potential energy is converted into kinetic energy. We can find the speed at point B using the formula for kinetic energy:
Kinetic energy at B = (1/2) * mass * speed^2
where mass = 0.050 kg and kinetic energy at B = 0.245 J.
Solving for speed^2:
0.245 J = (1/2) * 0.050 kg * speed^2
Rearranging the equation:
speed^2 = (2 * 0.245 J) / 0.050 kg
= 4.9 m^2/s^2
Taking the square root of both sides:
speed = √4.9 m^2/s^2
= 2.21 m/s
Therefore, the speed of the bead at point B is 2.21 m/s.
To find the speed of the bead at point B, we can use the principle of conservation of mechanical energy. At point A, the bead has gravitational potential energy due to its height, and when it reaches point B, this potential energy is converted into kinetic energy.
1. Start by finding the gravitational potential energy at point A:
Gravitational potential energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
Given: mass (m) = 50 g = 0.05 kg, height (h) = 0.5 m, acceleration due to gravity (g) = 9.8 m/s^2
PE at point A = 0.05 kg * 9.8 m/s^2 * 0.5 m = 0.245 J (Joules)
2. Since there is a friction force acting on the bead, there is work done against the friction force. This work reduces the total mechanical energy of the system and needs to be subtracted from the initial potential energy.
Work done against friction force (W) = friction force (F) * distance (d)
Given: friction force (F) = 0.020 N, distance (d) = 0.8 m
W = 0.020 N * 0.8 m = 0.016 J (Joules)
3. The total mechanical energy at point B is the sum of the remaining kinetic energy (KE) and the work done against friction:
Total mechanical energy at point B = KE + Work done against friction
Let's denote the final speed at point B as v.
KE at point B = 0.5 * mass (m) * velocity (v)^2
Total mechanical energy at point B = PE at point A - Work done against friction
PE at point A - Work done against friction = KE at point B
0.245 J - 0.016 J = 0.5 * 0.05 kg * v^2
4. Solve for v by rearranging the equation:
0.245 J - 0.016 J = 0.025 kg * v^2
0.229 J = 0.025 kg * v^2
v^2 = 0.229 J / 0.025 kg
v^2 = 9.16 m^2/s^2
v = √(9.16) m/s
v ≈ 3.03 m/s
Therefore, the speed of the bead at point B is approximately 3.03 m/s.
final KE=INitial PE - friction force
final KE=1/2 .050*.8-.020*.8
find velocity from the finalke