A small ball of mass 65 g is suspended from a
string of length 53 cm and whirled in a circle
lying in the horizontal plane.If the string makes an angle of 28◦ with the
vertical, find the centripetal force experienced
by the ball. The acceleration of gravity is
9.8 m/s
2
.
Answer in units of N
0.34 approx
mv^2 /r=Tsin28
mg=Tcos28
9circle is of radius r=.53sin28
To find the centripetal force experienced by the ball, we need to consider the forces acting on the ball.
The force acting on the ball in the vertical direction is the weight of the ball, which is given by:
Weight = mass * gravity
where the mass is given as 65 g (or 0.065 kg) and the acceleration due to gravity is given as 9.8 m/s^2.
Weight = 0.065 kg * 9.8 m/s^2 = 0.637 N
The force acting on the ball in the horizontal direction is the tension in the string. This tension provides the centripetal force that keeps the ball moving in a circle.
The tension in the string can be found using trigonometry. We can use the angle between the string and the vertical, which is given as 28 degrees, and the length of the string, which is given as 53 cm (or 0.53 m), to find the horizontal component of the tension.
Horizontal component of tension = tension * cos(angle)
Tension = Horizontal component of tension / cos(angle)
Tension = (Weight / cos(angle))
Substituting the values:
Tension = (0.637 N / cos(28 degrees))
Tension ≈ 0.723 N
Therefore, the centripetal force experienced by the ball is approximately 0.723 N.
To find the centripetal force experienced by the ball, we can use Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration.
In this case, the acceleration is the centripetal acceleration, which is given by the formula:
ac = (v^2) / r
Where:
ac = centripetal acceleration
v = velocity of the ball
r = radius of the circle (distance between the ball and the center of the circle)
To find the velocity of the ball, we can use the relationship between the velocity and the radius:
v = ωr
Where:
v = velocity
ω = angular velocity
The angular velocity (ω) can be found using the following formula:
ω = 2πf
Where:
f = frequency of rotation
To calculate the frequency, we can use the relationship between the frequency and the period (T) of rotation:
f = 1/T
Now, let's break down the information given in the question and solve step by step:
Mass of the ball (m) = 65 g = 0.065 kg
Length of the string (L) = 53 cm = 0.53 m
Angle made by the string with the vertical (θ) = 28 degrees
Acceleration due to gravity (g) = 9.8 m/s^2
Step 1: Calculate the radius (r) using the length of the string and the angle:
r = L * sin(θ)
r = 0.53 m * sin(28 degrees)
r ≈ 0.248 m
Step 2: Calculate the angular velocity (ω) using the frequency:
To find the frequency (f), we need to convert the period (T) to frequency using f = 1/T.
Let's assume the period T = 1 second.
f = 1/T = 1/1 = 1 Hz
Using the frequency, we can calculate the angular velocity:
ω = 2 * π * f
ω = 2 * π * 1
ω ≈ 6.28 rad/s
Step 3: Calculate the velocity (v):
v = ω * r
v = 6.28 rad/s * 0.248 m
v ≈ 1.555 m/s
Step 4: Calculate the centripetal acceleration (ac):
ac = (v^2) / r
ac = (1.555 m/s)^2 / 0.248 m
ac ≈ 9.777 m/s^2
Step 5: Calculate the centripetal force (F):
F = m * ac
F = 0.065 kg * 9.777 m/s^2
F ≈ 0.636 N
Therefore, the centripetal force experienced by the ball is approximately 0.636 N.