A purse at radius 1.70 m and a wallet at radius 2.50 m travel in uniform circular motion on the floor of a merry-go-round as the ride turns. They are on the same radial line. At one instant, the acceleration of the purse is (1.90 m/s2) + (3.00 m/s2). At that instant and in unit-vector notation, what is the acceleration of the wallet?
Well, isn't that quite the circus act! The purse and wallet seem to be having a wild time on the merry-go-round. Now, let's tackle this gravitational juggling act.
Since the purse and wallet are on the same radial line, we can safely assume that they have the same angular velocity (ω). This implies that their linear velocities (v) are directly proportional to their respective radii (r): v = ωr.
Now, let's get back to the accelerations. We are given that the acceleration of the purse is (1.90 m/s^2) + (3.00 m/s^2). However, this is not enough information to determine the acceleration of the wallet. We'll need to dig deeper into our bag of tricks to solve this one.
To find the acceleration of the wallet, we need to consider the net force acting on each object. Since their masses are not given, we'll assume they are negligible, and the only force acting on them is the tension in the radial line.
In circular motion, the centripetal force is given by F = m * a_c = m * v^2 / r, where a_c is the centripetal acceleration.
Since the objects are tied together, they experience the same centripetal force but in opposite directions. Therefore, the magnitudes of their centripetal accelerations are the same.
Now, at any instant, we have the following relation:
(mass of the purse) * (acceleration of the purse) = (mass of the wallet) * (acceleration of the wallet)
We don't know the masses of the purse and wallet, so we'll call them m_purse and m_wallet, respectively.
Using the centripetal acceleration formula, we have:
m_purse * a_purse = m_wallet * a_wallet,
But with the velocities being proportional to the radii, we can rewrite it as:
m_purse * (ω * r_purse) = m_wallet * (ω * r_wallet),
Cancel out the ω factor:
m_purse * r_purse = m_wallet * r_wallet,
Finally, we conclude that the ratios of the radii are equal to the ratios of their masses:
r_purse / m_purse = r_wallet / m_wallet.
So, unfortunately, without further information about the masses or the radius of the wallet, we cannot determine the acceleration of the wallet in unit-vector notation. It seems like the wallet is keeping its accelerational antics a secret for now!
To determine the acceleration of the wallet, we can use the concept of centripetal acceleration which is given by the formula:
a = ω² * r
where:
a is the centripetal acceleration,
ω (omega) is the angular velocity, and
r is the radius of the circular path.
In this case, we are given the centripetal acceleration of the purse, which is the same as the wallet since they are on the same radial line. We also know the radius of the circular path for both objects. However, we need to find the angular velocity.
To find the angular velocity, we can use the relationship between linear velocity (v) and angular velocity (ω):
v = ω * r
where:
v is the linear velocity, and
r is the radius of the circular path.
Given that both the purse and wallet are moving in uniform circular motion, their linear velocities are the same. Therefore, we can set up the following equation:
v_purse = v_wallet
ω * r_purse = ω * r_wallet
Since the angular velocities are the same, we can cancel them out:
r_purse = r_wallet
Now we can use this relationship to find the angular velocity of the purse:
a_purse = ω² * r_purse
Substituting the given values:
1.90 m/s² + 3.00 m/s² = ω² * 1.70 m
4.90 m/s² = ω² * 1.70 m
To isolate ω², divide both sides by 1.70 m:
ω² = 4.90 m/s² / 1.70 m
ω² = 2.88235 rad/s²
Now that we have the value of ω², we can use this to find the angular velocity of the wallet. Since the radii of the purse and wallet are the same:
ω_wallet = ω_purse
Taking the square root of ω²:
ω_wallet = √2.88235 rad/s²
Finally, to find the acceleration of the wallet, we can use the formula:
a_wallet = ω_wallet² * r_wallet
Substituting the known values:
a_wallet = (√2.88235 rad/s²)² * 2.50 m
a_wallet = 2.88235 rad/s² * 2.50 m
a_wallet = 7.2059 m/s²
So, the acceleration of the wallet, at the same instant, is (7.2059 m/s²) in unit-vector notation.