A 278 g particle is released from rest at point A inside an smooth hemispherical bowl of radius 36.5 cm, as shown below.
(a) Calculate the gravitational potential energy at A relative to B.
J
(b) Calculate the particle's kinetic energy at B.
J
(c) Calculate the particle's speed at B.
m/s
(d) Calculate the potential energy and kinetic energy at C.
PE = J
KE = J
To answer these questions, we need to apply the principles of conservation of energy. The total mechanical energy of the particle at any point (A, B, or C) is the sum of its gravitational potential energy and its kinetic energy.
(a) To calculate the gravitational potential energy at point A relative to B, we need to find the difference in height between A and B. Since the bowl is a hemisphere, the height is given by the radius, which is 36.5 cm. The formula for gravitational potential energy is PE = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height difference.
PE at A = m * g * h = 0.278 kg * 9.8 m/s^2 * 0.365 m = 0.993 J (rounded to three decimal places)
Therefore, the gravitational potential energy at A relative to B is approximately 0.993 J.
(b) The kinetic energy at B can be found using the principle of conservation of energy. The total mechanical energy at point A is equal to the total mechanical energy at point B (assuming no energy losses due to friction or other factors). Since the particle is released from rest at point A, its initial kinetic energy is zero. Therefore, the kinetic energy at B is equal to the gravitational potential energy at A.
KE at B = PE at A = 0.993 J
Therefore, the particle's kinetic energy at B is approximately 0.993 J.
(c) To calculate the particle's speed at B, we can use the formula for kinetic energy, KE = 1/2 mv^2, where m is the mass and v is the velocity (speed). Rearranging the formula, we get:
v^2 = 2 * KE / m
Plugging in the values, we have:
v^2 = 2 * 0.993 J / 0.278 kg
Simplifying, we find:
v^2 = 7.127 J/kg
Taking the square root of both sides, we get:
v ≈ √(7.127) = 2.67 m/s (rounded to two decimal places)
Therefore, the particle's speed at B is approximately 2.67 m/s.
(d) To find the potential energy and kinetic energy at C, we need to know the height difference between A and C. Unfortunately, this information is not provided. Without knowing the height difference, we cannot calculate the potential energy and kinetic energy at point C.