A 2.10-g particle is released from rest at point A on the inside of a smooth hemispherical bowl of radius R = 28.0 cm.
Calculate the following:
(a) Its gravitational potential energy at A relative to
J
(b) Its kinetic energy at B.
J
(c) Its speed at B.
m/s
(d) Its potential energy at C relative to B.
J
(e) Its kinetic energy at C.
J
A figure needs to be shown or described.
So does some effort on your part. This is a conservation of energy problem.
To calculate the values asked for in this problem, we need to use the concepts of gravitational potential energy and kinetic energy.
(a) To calculate the gravitational potential energy at point A, we can use the formula:
Potential Energy = mass × acceleration due to gravity × height
In this case, the height is the distance from point A to the bottom of the bowl, which is equal to the radius of the bowl. The mass of the particle is given as 2.10 grams (or 2.10 × 10^(-3) kg), and the acceleration due to gravity is approximately 9.8 m/s^2. Substituting these values into the formula, we get:
Potential Energy at A = 2.10 × 10^(-3) kg × 9.8 m/s^2 × 0.28 m
Calculating this expression gives us the answer in joules (J).
(b) The kinetic energy at point B can be calculated using the formula:
Kinetic Energy = (1/2) × mass × velocity^2
At point B, the particle will have fallen some distance from point A, and therefore gained some velocity. To find this velocity, we can consider the conservation of energy. The potential energy at A is converted into kinetic energy at B, so:
Potential Energy at A = Kinetic Energy at B
Using this, we can equate the expressions from part (a) and (b) and solve for the velocity.
(c) Once we have the velocity at point B, we can use it to calculate the speed at point B. Speed is simply the absolute value of the velocity, so we return the absolute value of the velocity we obtained in part (b).
(d) To calculate the potential energy at point C relative to point B, we use the same formula as in part (a), but with a different height. The height here is the additional distance the particle has fallen from point B to point C, which is equal to the radius of the bowl. Let's denote this additional height as Δh. Therefore, the potential energy at C relative to B is given by:
Potential Energy at C relative to B = mass × acceleration due to gravity × Δh
(e) Finally, the kinetic energy at point C can be calculated using the formula stated in part (b). The velocity at point C will be greater than at point B due to the increase in height, so the kinetic energy at C will be different.
Remember to always double-check your calculations and units when solving problems like this.