# Conservation of Energy with Mass on Hemisphere

• ccndy
In summary: And then you fall back to your original equation for the energy, ##1/2 m v^2 = mgh##, where ##h=R-Rcos(\theta)##. So, this yields the same answer as before, but with an initial velocity.
ccndy
Homework Statement
An object of mass m slides without friction on a hemispherical surface of radius, R. For t < 0, the
object is at rest at the very top of the sphere, i.e. θ = 0. At t = 0, it is given an instantaneous impulse, ∆~p = ∆pˆi. Determine the angle at which the mass leaves the hemisphere.
Relevant Equations
E = K + U
∆p = mv_f - mv_0
I tried approaching this question like this:

F_N - mgcos(theta) = -mR(theta_dot)^2

and theta_dot = v/R since R is constant

F_N = m(gcos(theta) - (v - v_0)^2/R) (with v being final velocity and v_0 being the initial velocity from the impulse)

and then using energy conservation:

at t = 0: E = 1/2(mv_0^2) + mgR
at t > 0: E = 1/2(mv^2) + mgRcos(theta)

Equating both equations, I got that:

(v - v_0)^2 = 2gR(1-cos(theta))

which would just yield the same angle as the situation where there is no impulse. Am I approaching the question wrong or am I incorporating impulse incorrectly?

Thanks.

ccndy said:
Homework Statement:: An object of mass m slides without friction on a hemispherical surface of radius, R. For t < 0, the
object is at rest at the very top of the sphere, i.e. θ = 0. At t = 0, it is given an instantaneous impulse, ∆~p = ∆pˆi. Determine the angle at which the mass leaves the hemisphere.
Relevant Equations:: E = K + U
∆p = mv_f - mv_0

I tried approaching this question like this:

F_N - mgcos(theta) = -mR(theta_dot)^2

and theta_dot = v/R since R is constant

F_N = m(gcos(theta) - (v - v_0)^2/R) (with v being final velocity and v_0 being the initial velocity from the impulse)

and then using energy conservation:

at t = 0: E = 1/2(mv_0^2) + mgR
at t > 0: E = 1/2(mv^2) + mgRcos(theta)

Equating both equations, I got that:

(v - v_0)^2 = 2gR(1-cos(theta))

which would just yield the same angle as the situation where there is no impulse. Am I approaching the question wrong or am I incorporating impulse incorrectly?

Thanks.

For example you have written:

F_N - mgcos(theta) = -mR(theta_dot)^2

In latex that is parsed as:

$$F_N - mg \cos( \theta) = -mR(\dot \theta)^2$$

ccndy said:
Homework Statement:: An object of mass m slides without friction on a hemispherical surface of radius, R. For t < 0, the
object is at rest at the very top of the sphere, i.e. θ = 0. At t = 0, it is given an instantaneous impulse, ∆~p = ∆pˆi. Determine the angle at which the mass leaves the hemisphere.
Relevant Equations:: E = K + U
∆p = mv_f - mv_0

which would just yield the same angle as the situation where there is no impulse. Am I approaching the question wrong or am I incorporating impulse incorrectly?
Why would it yield the same angle? Is the speed the same at a fixed angle with and without impulse at the top?

ccndy said:
Equating both equations, I got that:

(v - v_0)^2 = 2gR(1-cos(theta))
You have confused us by leaving out your final step, which would have produced gcos(theta)=2g(1-cos(theta)).
The equation I quote above is wrong. To get it you turned ##v^2-v_0^2## into ##(v-v_0)^2##.

haruspex said:
You have confused us by leaving out your final step, which would have produced gcos(theta)=2g(1-cos(theta)).
The equation I quote above is wrong. To get it you turned ##v^2-v_0^2## into ##(v-v_0)^2##.
You're right... that was a stupid algebraic mistake on my part.

However, does the incorporation of ##v_0## from the impulse make sense?

erobz said:

For example you have written:

F_N - mgcos(theta) = -mR(theta_dot)^2

In latex that is parsed as:

$$F_N - mg \cos( \theta) = -mR(\dot \theta)^2$$
Thank you!

SammyS and erobz
ccndy said:
You're right... that was a stupid algebraic mistake on my part.

However, does the incorporation of ##v_0## from the impulse make sense?
If it's given an instantaneous impulse such that the angle ##\theta## remains virtually ##0##, I would think yes...you basically start at ##t=0## with ##v_o## ( or ##\dot \theta_o## ).

Last edited:

## 1. What is the conservation of energy with mass on a hemisphere?

The conservation of energy with mass on a hemisphere refers to the principle that energy cannot be created or destroyed, but can only be transformed from one form to another. In this case, it specifically applies to the energy of a mass on a hemisphere, which remains constant as it moves and changes position.

## 2. How does the conservation of energy apply to a mass on a hemisphere?

The conservation of energy applies to a mass on a hemisphere because the mass has potential energy due to its position on the hemisphere, and this energy is constantly being converted into kinetic energy as the mass moves. The total energy of the mass remains the same, but it is continuously being transformed between potential and kinetic forms.

## 3. What factors affect the conservation of energy with mass on a hemisphere?

The conservation of energy with mass on a hemisphere can be affected by factors such as the mass of the object, the height of the hemisphere, and the speed at which the object is moving. These factors can impact the potential and kinetic energy of the mass and therefore affect the overall conservation of energy.

## 4. Why is the conservation of energy important in understanding the motion of a mass on a hemisphere?

The conservation of energy is important in understanding the motion of a mass on a hemisphere because it allows us to predict and explain the behavior of the mass. By understanding how energy is conserved and transformed, we can determine the speed, height, and other characteristics of the mass as it moves on the hemisphere.

## 5. How does the conservation of energy with mass on a hemisphere relate to other laws of physics?

The conservation of energy with mass on a hemisphere is closely related to other laws of physics, such as the law of conservation of momentum and the law of gravitation. These laws work together to explain the behavior of objects in motion and their interactions with each other. The conservation of energy is a fundamental principle that applies to all physical systems, including a mass on a hemisphere.

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