Physics - Calculating Potential Energy

A book falls off the top of a bookshelf. What is its potential energy right before it falls?

$\small g=-9.8\frac{m}{s^2}$

Explanation

The formula for potential energy is .

Given the values for the mass, height, and gravity, we can solve using multiplication. Note that the height is negative because the book falls in the downward direction.

$PE=3.23kg*-9.8\frac{m}{s^2}*-3.01m$

A pendulum with string length is dropped from rest. If the mass at the end of the pendulum is , what is its initial potential energy?

$\small g=-9.8\frac{m}{s^2}$

Explanation

Potential energy can be found using the equation . For the pendulum, the height is going to be the length of the string.

Remember, your height is your change in distance. In this case the ball will go down , so the height will be negative since the ball travels downward.

$PE= (2.03kg)(-9.8\frac{m}{s^2})(-1.2m)$

The units for energy are Joules.

Two balls, one with mass and one with mass , are dropped from above the ground. What is the potential energy of the ball right before it starts to fall?

$\small g=-9.8\frac{m}{s^2}$

Explanation

The equation for potential energy is .

Since we know the mass, height, and acceleration from gravity, we can simply multiply to find the potential energy.

$PE=(2kg)(-9.8\frac{m}{s^2})(-30m)$

Note that we plugged in for because the ball will be moving downward; the change in height is negative as the ball drops.

Two balls, one with mass and one with mass , are dropped from above the ground. What is the potential energy of the ball right before it starts to fall?

$\small g=-9.8\frac{m}{s^2}$

Explanation

The equation for potential energy is .

Since we know the mass, height, and acceleration from gravity, we can simply multiply to find the potential energy.

$PE=(4kg)(-9.8\frac{m}{s^2})(-30m)$

Note that we plugged in for because the ball will be moving downward; the change in height is negative as the ball drops.

A man stands at the top of a tall building. He holds a rock over the edge. What is the potential gravitational energy of the rock?

Explanation

Potential gravitational energy is given by the equation:

We are told the height of the rock and its mass. Using the constant acceleration due to gravity, we can solve for the gravitational potential energy.

$PE=(3.2kg)(9.8\frac{m}{s^2})(21m)$

$PE=658.56\ \frac{kg\cdot m^2}{s^2}=658.56J$

Laurence throws a rock off the edge of a tall building at an angle of $20^{\circ}$ from the horizontal with an initial speed of $31\frac{m}{s}$ .

$g=-9.8\frac{m}{s^2}$ .

What is the potential energy of the rock at the moment it is released?

Explanation

The formula for gravitational potential energy is:

We can solve for this value using the given mass of the rock, acceleration of gravity, and initial height.

$PE=(0.5kg)(9.8\frac{m}{s^2})(12m)$

This value is independent of the kinetic energy of the rock, and is not dependent on initial velocity.

A skier is at the top of a hill. At the bottom of the hill, she has a velocity of $\small 12\frac{m}{s}$ . How tall was the hill?

Explanation

At the top of the hill the skier has purely potential energy. At the bottom, she has purely kinetic energy.

We can solve by understanding the conservation of energy. The skier's energy at the top of the hill will be equal to her energy at the bottom of the hill.

$PE_{top}=KE_{bottom}$

Using the equations for potential and kinetic energy, we can solve for the height of the hill.

$mgh=\frac{1}{2}mv^2$

The masses cancel, and we can plug in our final velocity and gravitational acceleration.

$gh=\frac{1}{2}v^2$

$(-9.8\frac{m}{s^2})h=\frac{1}{2}(12\frac{m}{s})^2$

$(-9.8\frac{m}{s^2})h=\frac{1}{2}144\frac{m^2}{s^2}$

$(-9.8\frac{m}{s^2})h=72\frac{m^2}{s^2}$

$h=\frac{72\frac{m^2}{s^2}}{-9.8\frac{m}{s^2}}$

This formula solves for the change in height. The negative sign implies she travelled in a downward direction. Because the question is asking how tall the hill is, we use an absolute value.

An astronaut is on a new planet. She discovers that if she drops a space rock from above the ground, it has a final velocity of $3\frac{m}{s}$ just before it strikes the planet surface. What is the acceleration due to gravity on the planet?

$-0.45\frac{m}{s^2}$

$0.12\frac{m}{s^2}$

$-9.8\frac{m}{s^2}$

$-8.88\frac{m}{s^2}$

$-1.33\frac{m}{s^2}$

Explanation

We can use conservation of energy to solve. The potential energy when the astronaut is holding the rock will be equal to the kinetic energy just before it strikes the surface.

$mgh=\frac{1}{2}mv^2$

Now, we need to solve for $\small g$ , the gravity on the new planet. The masses will cancel out.

$gh=\frac{1}{2}v^2$

Plug in the given values and solve.

$g(-10m)=\frac{1}{2}(3\frac{m}{s})^2$

$g(-10m)=\frac{1}{2}(9\frac{m^2}{s^2})$

$g(-10m)=(4.5\frac{m^2}{s^2})$

$g=\frac{4.5\frac{m^2}{s^2}}{-10m}$

$g=-0.45\frac{m}{s^2}$

A ball rolls up a hill. If the ball is initially travelling with a velocity of $3.12\frac{m}{s}$ , how high up the hill does it roll?

$\small g=-9.8\frac{m}{s^2}$

Explanation

Use the conservation of energy equation to solve for the potential energy at the top of the hill.

$PE_{top}=KE_{bottom}$

$mgh=\frac{1}{2}mv^2$

Plug in the values given to you and solve for the final height.

$(0.5kg)(9.8\frac{m}{s^2})(h)=\frac{1}{2}(0.5kg)(3.12\frac{m}{s})^2$

$4.9\frac{kg\cdot m}{s^2}h=2.4336\frac{kg\cdot m^2}{s^2}$

$h=\frac{2.4336\frac{kg\cdot m^2}{s^2}}{4.9\frac{kg\cdot m}{s^2}}$

$h=0.497m\approx0.5m$

A book falls off the top of a bookshelf. What is its potential energy right before it falls?

$\small g=-9.8\frac{m}{s^2}$

Explanation

The formula for potential energy is .

Given the values for the mass, height, and gravity, we can solve using multiplication. Note that the height is negative because the book falls in the downward direction.

$PE=2kg*-9.8\frac{m}{s^2}*-2.3m$

Calculating Potential Energy

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