a dog sits 1.34m from the center of a merrygo round with an angular speed of 1.26rad/s. if the magnitude of the force that maintains the dog's circular motion is 43.8N, what is the dog's mass? answer in units of kg?
Solve
M R w^2 = 53.8 N, for M
M = 43.8/(R w^2)
w is the angular speed, 1.26 rad/s
R = 1.34 m
To find the dog's mass, we can use the centripetal force formula:
Fc = m * ω² * r
where Fc is the centripetal force, m is the mass of the dog, ω is the angular speed, and r is the distance from the center of the merry-go-round.
Given:
Fc = 43.8 N
ω = 1.26 rad/s
r = 1.34 m
Plugging in these values into the formula:
43.8 N = m * (1.26 rad/s)² * 1.34 m
Let's solve the equation step by step:
Square the angular speed:
(1.26 rad/s)² = 1.5876 rad²/s²
Multiply the squared angular speed by the radius:
1.5876 rad²/s² * 1.34 m = 2.126184 m³/s²
Now, divide both sides of the equation by 2.126184 m³/s²:
43.8 N / 2.126184 m³/s² = m
Calculating the result:
m ≈ 20.59 kg
Therefore, the dog's mass is approximately 20.59 kg.
To find the mass of the dog, we'll need to use the equation that relates the centripetal force, mass, and angular speed. The centripetal force acting on an object moving in a circle is given by the formula:
F = m * r * ω^2
Where:
F is the centripetal force,
m is the mass of the object,
r is the radius of the circular path,
and ω (omega) is the angular speed.
In this case, we are given the following information:
The centripetal force (F) is 43.8 N,
The radius (r) is 1.34 m,
and the angular speed (ω) is 1.26 rad/s.
Plugging the values into the formula, we get:
43.8 N = m * 1.34 m * (1.26 rad/s)^2
Now, let's solve for the mass (m):
Divide both sides of the equation by (1.34 m * (1.26 rad/s)^2):
m = 43.8 N / (1.34 m * (1.26 rad/s)^2)
Calculating this expression will give us the dog's mass:
m ≈ 22.18 kg
Therefore, the mass of the dog is approximately 22.18 kg.