A 32 g ball rolls around a 40 cm diameter L-shaped track, shown in the figure below, at 60 rpm. Rolling friction can be neglected. What is the magnitude of the net force exerted on the ball?
To find the magnitude of the net force exerted on the ball, we can use the following steps:
Step 1: Calculate the mass of the ball
Given that the ball has a mass of 32 g (grams), we need to convert it to kilograms (kg) since SI units are usually preferred in physics. 1 kg = 1000 g, so the mass of the ball becomes:
mass = 32 g = 32/1000 kg = 0.032 kg
Step 2: Calculate the linear velocity of the ball
The ball rolls around the track at 60 rpm (revolutions per minute). We need to convert this to meters per second (m/s) to match the SI units we will use. To do this, we can follow these steps:
- Convert 60 rpm to rev/s: 1 rev = 2π rad, and 1 min = 60 s, so 60 rpm = 60/60 rev/s = 1 rev/s.
- The circumference of the track can be calculated using the formula: circumference = π * diameter. Substituting the given diameter of 40 cm = 0.4 m, we get: circumference = π * 0.4 m = 1.26 m.
- The linear velocity (v) of the ball can be found by multiplying the circumference by the number of revolutions per second: v = 1.26 m/rev * 1 rev/s = 1.26 m/s.
Step 3: Calculate the centripetal force
The ball moves in a circular path, so it experiences a centripetal force (Fc). The centripetal force can be calculated using the formula: Fc = m * (v^2 / r), where m is the mass, v is the linear velocity, and r is the radius of the circular path.
- The radius (r) of the circular path is equal to half the diameter of the track: r = 0.4 m / 2 = 0.2 m.
- Now we can plug in the values:
Fc = 0.032 kg * (1.26 m/s)^2 / 0.2 m = 0.5088 kg·m/s^2 (which is equivalent to a Newton, N).
Since no other forces are mentioned in the problem, this is the magnitude of the net force exerted on the ball. Therefore, the magnitude of the net force is 0.5088 N.
To find the magnitude of the net force exerted on the ball, we need to consider the centripetal force acting on the ball as it rolls around the L-shaped track.
First, let's calculate the angular velocity of the ball in radians per second.
Angular velocity (ω) is given by the formula:
ω = 2πf,
where f is the frequency in hertz (Hz) or revolutions per second (rps).
In this case, the ball rotates at 60 revolutions per minute (rpm), so the frequency (f) can be calculated as:
f = 60 rpm / 60 seconds = 1 rps.
Now we can calculate the angular velocity:
ω = 2π(1 rps) = 2π rad/s.
Next, we need to calculate the linear velocity (v) of the ball as it rolls around the track. The linear velocity can be calculated using the formula:
v = ωr,
where r is the radius of the circular path.
The radius (r) of the track is half of the diameter (d), so:
r = 40 cm / 2 = 20 cm = 0.20 m.
Plugging in the values, we get:
v = (2π rad/s)(0.20 m) = 1.26 m/s.
Now, let's calculate the centripetal force (F) acting on the ball using the formula:
F = mv² / r,
where m is the mass of the ball.
The mass (m) is given as 32 g, so we need to convert it to kilograms:
m = 32 g / 1000 = 0.032 kg.
Plugging in the values, we get:
F = (0.032 kg)(1.26 m/s)² / 0.20 m = 0.807 N.
Therefore, the magnitude of the net force exerted on the ball is approximately 0.807 N.