Let's break down and solve each problem.

1. Potential energy of a lifted box

The formula for gravitational potential energy is: $PE = mgh$ Where:

• $PE = 176 \, \text{J}$ (given),
• $m = 6.2 \, \text{kg}$ (mass of the box),
• $g = 9.8 \, \text{m/s}^2$ (acceleration due to gravity),
• $h$ is the height of the shelf.

We rearrange the equation to solve for $h$: $h = \frac{PE}{mg} = \frac{176}{6.2 \times 9.8}$ $h = \frac{176}{60.76} \approx 2.9 \, \text{m}$ So, the shelf is approximately 2.9 meters high.

2. Mass of the man climbing a wall

Using the same potential energy formula: $PE = mgh$ Where:

• $PE = 3168 \, \text{J}$,
• $h = 5.2 \, \text{m}$,
• $g = 9.8 \, \text{m/s}^2$.

Rearranging to solve for $m$: $m = \frac{PE}{gh} = \frac{3168}{9.8 \times 5.2}$ $m = \frac{3168}{50.96} \approx 62.2 \, \text{kg}$ So, the mass of the man is approximately 62.2 kg.

3. Potential energy of the ball on a slope

First, we need to calculate the vertical height of the ball using trigonometry. The right triangle has:

• Opposite side (height) = 20 cm = 0.2 m,
• Hypotenuse = 50 cm = 0.5 m.

The angle $\theta$ can be calculated by: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{0.2}{0.5} = 0.4$ Now, using potential energy: $PE = mgh$ Where:

• $m = 907 \, \text{g} = 0.907 \, \text{kg}$,
• $g = 9.8 \, \text{m/s}^2$,
• $h = 0.2 \, \text{m}$.

$PE = 0.907 \times 9.8 \times 0.2 \approx 1.78 \, \text{J}$ The ball gains approximately 1.78 J of potential energy.

4. Kinetic energy of a moving car

The formula for kinetic energy is: $KE = \frac{1}{2} mv^2$ Where:

• $m = 805 \, \text{kg}$,
• $v = 26 \, \text{m/s}$.

$KE = \frac{1}{2} \times 805 \times 26^2 = \frac{1}{2} \times 805 \times 676 = 272,690 \, \text{J}$ So, the kinetic energy of the car is approximately 272,690 J (or 272.69 kJ).

5. Kinetic energy of a cyclist and bike

Using the same kinetic energy formula: $KE = \frac{1}{2} mv^2$ Where:

• $m = 111 \, \text{kg}$ (total mass of cyclist and bike),
• $v = 12 \, \text{m/s}$.

$KE = \frac{1}{2} \times 111 \times 12^2 = \frac{1}{2} \times 111 \times 144 = 7,992 \, \text{J}$ The kinetic energy is approximately 7,992 J (or 7.99 kJ).

6. Mass of a tennis ball based on kinetic energy

Rearrange the kinetic energy formula to solve for mass: $KE = \frac{1}{2} mv^2 \quad \text{so} \quad m = \frac{2KE}{v^2}$ Where:

• $KE = 72 \, \text{J}$,
• $v = 53 \, \text{m/s}$.

$m = \frac{2 \times 72}{53^2} = \frac{144}{2809} \approx 0.0512 \, \text{kg}$ The mass of the tennis ball is approximately 0.0512 kg (or 51.2 g).

Would you like further details on any of these solutions?

Here are 5 follow-up questions related to these problems:

1. How would the potential energy change if the box's mass doubled in problem 1?
2. What effect does increasing the height of the wall in problem 2 have on potential energy?
3. How does the angle of the slope affect the potential energy gain in problem 3?
4. If the car's speed was doubled, how would its kinetic energy change?
5. How does the kinetic energy of the cyclist and bike change if they slow down to 6 m/s?

Tip: Always keep track of units in physics problems to avoid calculation errors.