A bead slides without friction around a loop–the–loop (see figure below). The bead is released from rest at a height h = 3.60R.
(a) What is its speed at point ? (Use the following as necessary: the acceleration due to gravity g, and R.)
v =
So I can't get this. I know the form of a would be sqrt((number)*g*R) but how do you get the number?
An electric scooter has a battery capable of supplying 130 Wh of energy. If friction forces and other losses account for 60.0% of the energy usage, what altitude change can a rider achieve when driving in hilly terrain, if the rider and scooter have a combined weight of 820 N?
i got this to be 342.44 by doing .6*130*3600=280800J then 280800=h*820. whats wrong in this case?
To find the speed at point B in the loop-the-loop, you need to apply the conservation of mechanical energy. At point B, the total mechanical energy (kinetic energy + potential energy) is equal to the potential energy at the starting point A.
1. Determine the potential energy at point A:
The potential energy at point A is given by potential energy = mass * gravity * height. Since the bead has no mass specified, we can cancel it out from both sides of the equation.
Potential energy at A: PE_a = mass * gravity * height = mgh = gh, where h is the height above the starting point and R is the radius of the loop.
2. Apply conservation of mechanical energy:
At point B, the total mechanical energy is equal to the potential energy at A:
Total mechanical energy at B = Potential energy at A
Total mechanical energy at B = kinetic energy at B + potential energy at B
Since the bead is released from rest, the kinetic energy at B is initially zero.
Total mechanical energy at B = 0 + potential energy at B
Potential energy at B can be calculated using the same formula as in step 1, with the height at B being R (lowermost point of the loop):
Potential energy at B: PE_b = gh
Setting the total mechanical energies at B and A equal to each other:
gh = gh + PE_b
3. Solve for speed at point B:
At point B, the total energy is given by kinetic energy, which is equal to half the mass times the square of the speed (0.5 * mass * velocity^2).
0.5 * mv^2 = gh + PE_b
Solving for velocity:
v^2 = 2gR
Taking the square root of both sides:
v = √(2gR)
Substituting in the given values for h = 3.60R:
v = √(2gR) = √(2g(3.60R))
Now you can plug in the known values of g and R to calculate the speed at point B.
Regarding the second question, you made a small error in your calculations. The correct formula for the energy available for altitude change should be:
Available energy = total energy - energy losses
Available energy = 130 Wh - (0.6 * 130 Wh)
Note that the energy losses are subtracted because they are not available for altitude change.
Then, convert the available energy from watt-hours (Wh) to joules (J) by multiplying by 3600 (since 1 Wh = 3600 J):
Available energy (J) = (130 Wh - (0.6 * 130 Wh)) * 3600
Finally, you can divide the available energy by the weight to find the altitude change:
Altitude change = Available energy (J) / weight (N)
Substituting the given values:
Altitude change = ((130 Wh - (0.6 * 130 Wh)) * 3600) / 820
Now you can calculate the altitude change using these steps and values.