A coin with a diameter of 1.9 cm is dropped on edge onto a horizontal surface. The coin starts out with an initial angular speed of 19 rad/s and rolls in a straight line without slipping. The rotation slows with an angular acceleration of magnitude 2.1 rad/s2.
What is the initial linear velocity of the coin i got the acceleration as 1.80×10-1 m/s
How do you figure out
What is the magnitude of the linear acceleration
How far does the coin roll before it stops?
To find the initial linear velocity of the coin, you can use the formula for linear velocity in terms of angular velocity and radius:
Linear velocity (v) = Angular velocity (ω) * Radius (r)
Given that the diameter (d) of the coin is 1.9 cm, the radius (r) would be half of the diameter, so r = 0.95 cm = 0.0095 m. The initial angular velocity (ω) is 19 rad/s.
v = 19 rad/s * 0.0095 m
v ≈ 0.1805 m/s
Therefore, the initial linear velocity of the coin is approximately 0.1805 m/s.
To find the magnitude of the linear acceleration, you can use the formula for linear acceleration in terms of angular acceleration and radius:
Linear acceleration (a) = Angular acceleration (α) * Radius (r)
Given that the magnitude of the angular acceleration (α) is 2.1 rad/s² and the radius (r) is 0.0095 m, we can plug the values into the formula:
a = 2.1 rad/s² * 0.0095 m
a ≈ 0.01995 m/s²
Therefore, the magnitude of the linear acceleration is approximately 0.01995 m/s².
To find how far the coin rolls before it stops, you can use the equation for angular displacement:
Angular displacement (θ) = (ω² - ω₀²) / (2 * α)
Where ω is the final angular velocity (which will be zero, as the coin stops rolling), ω₀ is the initial angular velocity, and α is the magnitude of the angular acceleration.
θ = (0 - (19 rad/s)²) / (2 * 2.1 rad/s²)
θ = -361 / 8.4
θ ≈ -42.9762 rad
Since the coin rolls in a straight line without slipping, the distance it travels is equal to the arc length of the circular path that the coin traces. We can use the formula for arc length to find this distance:
Arc length (s) = θ * r
s = (-42.9762 rad) * 0.0095 m
s ≈ -0.4083 m
However, since distance can't be negative, the coin rolls approximately 0.4083 m before it stops.
To find the initial linear velocity of the coin, you need to use the relationship between linear and angular velocity. When a coin rolls without slipping, the linear velocity of a point on the edge is equal to the product of the angular velocity and the radius of the coin.
Given:
Diameter of the coin = 1.9 cm = 0.019 m
Radius of the coin = (0.019 m) / 2 = 0.0095 m
Initial angular speed = 19 rad/s
To find the initial linear velocity, you can use the formula:
Linear velocity = Angular velocity * Radius
Substituting the given values into the formula:
Linear velocity = 19 rad/s * 0.0095 m
Linear velocity = 0.1805 m/s
Therefore, the initial linear velocity of the coin is 0.1805 m/s.
To find the magnitude of the linear acceleration, you can use the relationship between linear and angular acceleration. When an object rotates without slipping, the linear acceleration of a point on the edge is equal to the product of the angular acceleration and the radius of the coin.
Given:
Angular acceleration = 2.1 rad/s^2
Radius of the coin = 0.0095 m
To find the magnitude of the linear acceleration, you can use the formula:
Linear acceleration = Angular acceleration * Radius
Substituting the given values into the formula:
Linear acceleration = 2.1 rad/s^2 * 0.0095 m
Linear acceleration = 0.01995 m/s^2
Therefore, the magnitude of the linear acceleration is 0.01995 m/s^2.
To determine how far the coin rolls before it stops, you need to use the concept of rotational kinetic energy. The final energy of the rolling coin is zero, so the initial rotational kinetic energy must dissipate through work done against friction. The work done by friction is equal to the initial rotational kinetic energy.
The initial rotational kinetic energy is given by the formula:
Rotational Kinetic Energy = (1/2) * Moment of Inertia * (Angular velocity)^2
The moment of inertia for a solid cylinder, such as a coin, is (1/2) * Mass * Radius^2. Let's assume a mass for the coin.
Given:
Angular acceleration = -2.1 rad/s^2 (negative sign denotes deceleration)
Radius of the coin = 0.0095 m
To find the distance rolled by the coin, you can use the formula:
Distance = (Initial Angular velocity)^2 / (2 * |Angular acceleration|)
Substituting the given values into the formula:
Distance = (19 rad/s)^2 / (2 * 2.1 rad/s^2)
Distance = 19^2 / (2 * 2.1)
Distance = 1711 / 4.2
Distance ≈ 407.38 meters
Therefore, the coin rolls approximately 407.38 meters before it stops.