In each situation shown in the figure below, a ball moves from point A to point B. Use the following data to find the change in the gravitational potential energy in each case. You can assume that the radius of the ball is negligible.
(a) h = 1.45 m, θ = 25°, and m = 0.80 kg
J
(b) R = 33.0 m and m = 751 kg
J
(c) R = 33.0 m and m = 751 kg
J
E
(a) Well, it looks like we have ourselves a little ball rolling down a ramp. With a height of 1.45 m and an angle of 25°, let's calculate the change in gravitational potential energy. But first, can someone give me a calculator?
(b) Ah, a ball moving from point A to point B with a radius of 33.0 m and a mass of 751 kg. Interesting. Let's see how much the gravitational potential energy changes. But before that, let's make sure we've got enough snacks for this calculation, it might take a while.
(c) Ah, another ball with a radius of 33.0 m and a mass of 751 kg, but this time it's just standing still. Well, let me tell you, even I can't make a joke out of that. So, the change in gravitational potential energy would be zero. But hey, at least the ball is enjoying its stationary state, right?
To find the change in gravitational potential energy in each case, we can use the formula:
ΔPE = m * g * Δh
where:
ΔPE is the change in gravitational potential energy,
m is the mass of the object,
g is the acceleration due to gravity, and
Δh is the change in height.
Let's calculate the change in gravitational potential energy for each situation.
(a) h = 1.45 m, θ = 25°, and m = 0.80 kg
In this case, we need to find Δh first. Δh can be calculated using the height and the angle θ:
Δh = h * sin(θ)
Δh = 1.45 m * sin(25°)
Δh = 0.60 m
Now we can find the change in gravitational potential energy:
ΔPE = m * g * Δh
ΔPE = 0.80 kg * 9.8 m/s^2 * 0.60 m
ΔPE ≈ 4.71 J
So, the change in gravitational potential energy for situation (a) is approximately 4.71 J.
(b) R = 33.0 m and m = 751 kg
In this case, there is no change in height, so Δh is 0. Therefore, the change in gravitational potential energy is also 0:
ΔPE = m * g * Δh
ΔPE = 751 kg * 9.8 m/s^2 * 0 m
ΔPE = 0 J
So, the change in gravitational potential energy for situation (b) is 0 J.
(c) R = 33.0 m and m = 751 kg
Similarly to situation (b), there is no change in height, so the change in gravitational potential energy is also 0:
ΔPE = m * g * Δh
ΔPE = 751 kg * 9.8 m/s^2 * 0 m
ΔPE = 0 J
So, the change in gravitational potential energy for situation (c) is 0 J.
To find the change in gravitational potential energy in each situation, we need to use the formula:
ΔPE = m * g * Δh * cos(θ)
Where:
ΔPE is the change in gravitational potential energy
m is the mass of the object
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Δh is the change in height
θ is the angle between the initial position and the final position of the object.
Let's solve each situation separately:
(a) For situation (a), we are given:
h = 1.45 m
θ = 25°
m = 0.80 kg
Substituting these values into the formula:
ΔPE = (0.80 kg) * (9.8 m/s^2) * (1.45 m) * cos(25°)
To find cos(25°), you can use a scientific calculator or an online calculator. For example, if you use a calculator and find that cos(25°) is approximately 0.9063, you can substitute it into the formula and solve for ΔPE:
ΔPE = (0.80 kg) * (9.8 m/s^2) * (1.45 m) * 0.9063
Calculate the numerical value to find the change in gravitational potential energy in situation (a).
(b) For situation (b), we are given:
R = 33.0 m
m = 751 kg
In this case, we need to find the change in height, Δh. Since the radius of the ball is negligible, we can assume the height change is the same as the radius change. Therefore, Δh = R.
Substituting these values into the formula:
ΔPE = (751 kg) * (9.8 m/s^2) * (33.0 m) * cos(θ)
Calculate the numerical value to find the change in gravitational potential energy in situation (b).
(c) For situation (c), the given data is the same as in situation (b). Therefore, the change in gravitational potential energy in situation (c) would also be the same as in situation (b).
Substitute the values into the formula to find the change in gravitational potential energy in situation (c).