A 10 g particle has the potential energy shown in a graph with the y-axis label U(J) and the x axis is labeled in cm. There is a diagonal line from 4 J down to 1 cm and the line levels out on the x axis to 3 cm and rises again to a height of 2 J at 5 cm and levels out to 7 cm and rises to 6 J at 8 cm.
Part A: How much work does the force do as the particle moves from 2 cm to 6 cm?
W = ? J
Part B: What speed does the particle need at x = 2cm to arrive at x = 6m with a speed of 10 m/s ?
v = ? m/s
see the above question and response, same technique.
What formula do I use to find the work in part A?
I cannot figure part b which should be 6 cm (I typed m in original problem) Please help!!
To answer these questions, we need to understand the relationship between potential energy and work, as well as the relationship between work, force, and speed.
Part A: How much work does the force do as the particle moves from 2 cm to 6 cm?
Work is defined as the change in potential energy of an object, and it can be calculated using the equation:
W = ΔU
where W is the work done, and ΔU is the change in potential energy.
To find the change in potential energy, we need to determine the difference in potential energy between the initial and final positions of the particle.
1. Identify the initial and final positions of the particle. In this case, the initial position is 2 cm, and the final position is 6 cm.
2. Determine the potential energy at both positions. By examining the graph, we can see that the potential energy at 2 cm is 3 J, and the potential energy at 6 cm is 6 J.
3. Calculate the change in potential energy by subtracting the initial potential energy from the final potential energy: ΔU = 6 J - 3 J = 3 J.
4. Therefore, the work done by the force as the particle moves from 2 cm to 6 cm is 3 J.
Part A: W = 3 J
Part B: What speed does the particle need at x = 2 cm to arrive at x = 6 cm with a speed of 10 m/s?
To determine the speed of the particle at x = 2 cm, we can use the principle of conservation of energy. The initial potential energy and the initial kinetic energy of the particle must be equal to the final potential energy and the final kinetic energy.
1. Identify the initial and final positions of the particle. In this case, the initial position is 2 cm, and the final position is 6 cm.
2. Determine the potential energy at both positions. By examining the graph, we can see that the potential energy at 2 cm is 3 J, and the potential energy at 6 cm is 6 J.
3. Calculate the initial potential energy: U(initial) = 3 J.
4. Calculate the final potential energy: U(final) = 6 J.
5. To calculate the initial kinetic energy, we subtract the initial potential energy from the total mechanical energy at the initial position. Since the only energy present at the initial position is potential energy, the initial kinetic energy is 0 J.
6. Calculate the final kinetic energy by subtracting the final potential energy from the total mechanical energy at the final position. Since the total mechanical energy is the sum of potential and kinetic energy, we can determine the final kinetic energy using the equation: K(final) = [Total Energy - U(final)] = [3 J + K(initial) - 6 J].
7. We know that the final speed is 10 m/s. The kinetic energy of a particle is given by the equation: K = (1/2)mv^2, where m is the mass of the particle and v is its speed.
8. Rearrange the equation to solve for the initial kinetic energy: K(initial) = K(final) - (1/2)mv^2.
9. Substitute the known values: K(initial) = 3 J + K(initial) - 6 J - (1/2)(10 g)(10 m/s)^2.
10. Solve for K(initial) by rearranging the equation: K(initial) - K(initial) = -3 J - (1/2)(10 g)(10 m/s)^2.
11. Calculate the mass value in kg: 10 g = 0.01 kg.
12. Simplify the equation: 0 = -3 J - (1/2)(0.01 kg)(10 m/s)^2.
13. Solve for K(initial): K(initial) = -3 J - (1/2)(0.01 kg)(10 m/s)^2.
14. Calculate the value to find the speed at x = 2 cm: v^2 = 2K/0.01 kg.
15. Substitute the known values and calculate the speed: v^2 = 2(-3 J)/0.01 kg.
16. Solve for v: v = square root((-6 J)/(0.01 kg)).
17. Simplify the equation: v = square root(-600 m^2/s^2).
18. Since the speed cannot be imaginary, we conclude that the particle cannot have a speed of 10 m/s when it reaches x = 6 cm, starting from x = 2 cm.