A 245 g particle is released from rest at point A inside a smooth hemispherical bowl of radius 35.0 cm, as shown in Figure 5-21.
(a) Calculate the gravitational potential energy at A relative to B.
1 .
(b) Calculate the particle's kinetic energy at B.
2
(c) Calculate the particle's speed at B.
3
(d) Calculate the kinetic energy and potential energy at C.
4 (kinetic energy)
5 (potential energy)
To answer these questions, let's break down the problem step by step:
(a) To calculate the gravitational potential energy at point A relative to point B, we need to use the equation for gravitational potential energy:
PE = mgh
where m is the mass of the particle (245 g = 0.245 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height difference between points A and B. In this case, since the particle is released from rest, we can assume that the initial height is zero at A, and the final height at B is the height of the hemisphere's rim, which is equal to the radius of the hemisphere (35.0 cm = 0.35 m).
So, the gravitational potential energy at A relative to B is:
PE_AB = m * g * h
= 0.245 kg * 9.8 m/s^2 * 0.35 m
= 0.08421 J
(b) To calculate the particle's kinetic energy at point B, we need to use the equation for kinetic energy:
KE = (1/2) * m * v^2
where m is the mass of the particle (0.245 kg) and v is the velocity of the particle at point B. Since the particle is released from rest at point A, it will gain speed as it falls due to the gravitational force. At point B, the particle would have converted all its potential energy to kinetic energy.
To find the velocity at point B, we can use the conservation of energy principle. The potential energy at A, which is equal to the gravitational potential energy at A relative to B, is transformed into kinetic energy at point B.
PE_A = KE_B
So, the particle's kinetic energy at point B is:
KE_B = PE_AB
= 0.08421 J
(c) To calculate the particle's speed at point B, we can use the equation for kinetic energy and rearrange it to solve for velocity:
KE = (1/2) * m * v^2
Rearranging,
v^2 = (2 * KE) / m
Plugging in the values,
v^2 = (2 * 0.08421 J) / 0.245 kg
= 0.68865 m^2/s^2
Therefore, the particle's speed at point B is:
v = √(0.68865 m^2/s^2)
= 0.8297 m/s
(d) To calculate the kinetic energy and potential energy at point C, we need to know the height difference between points B and C. Without this information, we cannot determine the exact values of kinetic energy and potential energy at point C.