Juggles the clown stands on one end of a teeter-totter at rest on the ground. Bangles the clown jumps off a platform 2.5 m above the ground and lands on the other end of the teeter-totter, launching Juggles into the air. Juggles rises to a height of 3.7 m above the ground, at which point he has the same amount of gravitational potential energy as Bangles had before he jumped, assuming both potential energies are measured using the ground as the reference level. Bangles' mass is 75 kg. What is Juggles' mass?
Assuming no loss in the impact(s), then, let
M=mass of Juggles
and equate gravitational potential energy (g=acceleration due to gravity)
75kg*g*2.5=M*3.7*g
Solve for M.
To solve this problem, we can use the concept of conservation of energy.
1. First, let's find the gravitational potential energy of Bangles before he jumps. The formula for gravitational potential energy is:
Potential energy = mass * gravity * height
Given information:
Mass of Bangles (m1) = 75 kg
Height (h1) = 2.5 m
Gravity (g) = 9.8 m/s^2
Potential energy of Bangles:
Potential energy of Bangles (PE1) = m1 * g * h1
2. Now, let's find the gravitational potential energy of Juggles at his maximum height. The formula is the same as above:
Potential energy = mass * gravity * height
Given information:
Height of Juggles (h2) = 3.7 m
Gravity (g) = 9.8 m/s^2
Potential energy of Juggles:
Potential energy of Juggles (PE2) = m2 * g * h2
3. According to the problem, PE1 = PE2. Therefore, we can equate the two expressions and solve for m2, which is Juggles' mass.
m1 * g * h1 = m2 * g * h2
Cancelling out the 'g' and solving for m2:
m2 = (m1 * h1) / h2
Substituting the given values:
m2 = (75 kg * 2.5 m) / 3.7 m
Calculating:
m2 = 50.676 kg
Therefore, Juggles' mass is approximately 50.68 kg.
To solve this problem, we can use the conservation of momentum principle. According to this principle, the total momentum before and after the interaction remains constant. In this case, the interaction involves Bangles jumping and Juggles being launched into the air.
We can start by determining the initial momentum of Bangles before he jumps.
The equation for momentum is given as:
momentum = mass × velocity
Since there is no information given about Bangles' initial velocity, we can assume it is zero, as stated in the problem that the teeter-totter is initially at rest. Therefore, the initial momentum of Bangles is:
momentum_initial = mass × velocity_initial
= 75 kg × 0
= 0 kg·m/s
According to the conservation of momentum, the total momentum before the interaction is equal to the total momentum after the interaction. Thus, the momentum of Juggles after being launched into the air would be equal in magnitude but opposite in direction to Bangles' initial momentum.
momentum_final = -momentum_initial
Now we can find the momentum of Juggles after being launched into the air. Since his momentum is opposite in direction to Bangles' initial momentum and equal in magnitude, we can write:
momentum_final = -momentum_initial
= -(0 kg·m/s)
= 0 kg·m/s
Since momentum is given by:
momentum = mass × velocity
And we know that the momentum is zero, we can conclude that Juggles' final velocity is also zero (since mass cannot be zero). Therefore, after reaching a height of 3.7 m, Juggles is momentarily at rest in mid-air.
Now let's consider the gravitational potential energy. The equation for gravitational potential energy is:
gravitational potential energy = mass × acceleration due to gravity × height
Using the given values, we have:
Bangles' gravitational potential energy = 75 kg × 9.8 m/s^2 × 2.5 m
= 1837.5 J
According to the problem, Juggles' gravitational potential energy is also 1837.5 J when he reaches a height of 3.7 m.
Now we can calculate Juggles' mass. Rearranging the equation for gravitational potential energy:
mass = gravitational potential energy / (acceleration due to gravity × height)
mass = 1837.5 J / (9.8 m/s^2 × 3.7 m)
≈ 50 kg
Therefore, Juggles' mass is approximately 50 kg.